A Protestant web site has posted "Top 10 Reasons why Peter isn't the Rock on which Christ Jesus built His Church," challenging Catholics to explain these 10 objections. Gladly does the Catholic Church provide 10 answers. My responses are as follows (please read through all responses because some later responses clarify earlier responses):

Objection 1:
"In Greek, Peter is Petros, it is masculine in form, while, also in Greek petra or rock is feminine, i.e., it is more plausible that Jesus is referring to anything other than Peter."

Answer:
No, that is clearly absurd, for several reasons. Consider the context again:

St. Matthew 16:13-19
And Jesus came into the quarters of Cesarea Philippi: and he asked his disciples, saying: Whom do men say that the Son of man is? But they said: Some John the Baptist, and other some Elias, and others Jeremias, or one of the prophets. Jesus saith to them: But whom do you say that I am? Simon Peter answered and said: Thou art Christ, the Son of the living God. And Jesus answering said to him: Blessed art thou, Simon Bar-Jona: because flesh and blood hath not revealed it to thee, but my Father who is in heaven. And I say to thee: That thou art Peter; and upon this rock I will build my church, and the gates of hell shall not prevail against it. And I will give to thee the keys of the kingdom of heaven. And whatsoever thou shalt bind upon earth, it shall be bound also in heaven: and whatsoever thou shalt loose on earth, it shall be loosed also in heaven.

Note that this is the passage in which Simon's name is changed to Petros (in Greek) or Cephas (in Aramaic; cf. St. John 1:42). Cephas means rock; there is no dispute about that. It is only a matter of common sense that Petros also means rock, even though the literal word is petra, which is a feminine noun, and it is really not difficult to imagine why Christ the Lord did not wish to give Simon a feminine name! Secondly, if you look at the above context, if "upon this rock" does not refer to St. Peter, then why did Christ change Simon's name right then and there? What was the purpose of adding "Thou art Peter"? Could Christ not have changed that name at another occasion? Why would the Lord effectively say, "Blessed are you, Simon. You are now to be called Peter, and upon <anything else> I now build my Church. Here are the keys to the Kingdom of Heaven!" ??

Thirdly, our English language obscures the pun being made in this verse, but it's clear in some of the other languages of the world (I think we need to remind ourselves every now and then that English isn't the only language in the world). So, here's a little game now for the readers: read the verse (St. Matthew 16:18) in the following different languages and see if you can pick out which two words in each of these examples are the same (or very close to being the same).

GREEK: kago de soi lego hoti su ei Petros, kai epi taute te petra oikodomeso mou ten ekklesian, kai pulai hadou ou katischusousin autes.

FRENCH: Et moi, je te déclare : Tu es Pierre, et sur cette pierre j'édifierai mon Eglise, contre laquelle la mort elle-même ne pourra rien.

ITALIAN: E io ti dico: Tu sei Pietro e su questa pietra edificherò la mia chiesa e le porte degli inferi non prevarranno contro di essa.

LATIN: et ego dico tibi quia tu es Petrus et super hanc petram aedificabo ecclesiam meam et portae inferi non praevalebunt adversum eam.

SPANISH: Yo te digo que tú eres Pedro, y sobre esta piedra edificaré mi iglesia, y las puertas del reino de la muerte no prevalecerán contra ella

Even linguistically, you can see that these two words are related to each other, and refer to each other in this sentence. Just by looking at the other languages, it should be clear on the face of it, especially in the French! While the author says, "it is more plausible that Jesus is referring to anything other than Peter," what is obvious just by looking at the words themselves is that it is highly unlikely that "petra" refers to anything except "Petros."

Objection 2:
"Jesus used a personal pronoun addressing Petros (i.e., "YOU are Peter") but regarding the petra, Jesus used demonstrative pronoun (i.e., "UPON this rock"), in third person."

Answer:
 

So what? How does that disprove the Catholic claim? Christ was addressing Simon directly, so He said, "You are [from now on] Peter." He then used the demonstrative pronoun "this" to emphasize that He was building the Church on this very rock that is Peter, not on any of the other disciples (cf. St. Matthew 16:13). In other words, He said, "You are Peter, and upon this rock that you are, I will build my Church." This becomes even clearer if we use "rock" instead of "Peter": "You are the rock, and upon this rock I will build my Church." What's so strange about that? Christ used the demonstrative pronoun "this" because he was using a metphor ("rock") that applies to Peter. Look, Christ did not merely give Simon a new name, he also made clear at the same time why Simon was receiving that new name, Petros. And the reason is that Simon is now "this rock [upon which] I will build my Church"!

Objection 3:
"If Jesus meant Petros to be the petra, there is no reason why he shouldn't have said, "UPON YOU I will build my Church," but he didn't."

Answer:
 

He could have, but the effect would not have been the same. Hello! The whole point was to change Simon's name into Peter, and by saying "upon this rock" (as opposed to "upon you") Christ made clear why the name was being changed: because Peter was now "this rock" upon which the Church would be built! The very fact that Christ said "upon this rock" and not "upon you" actually lends further credibility to the Catholic assertion! Interestingly enough, the Protestant author here has just refuted his own "Objection 5" (see below), in which he says: "It is actually more plausible, based on the context, that when Jesus said, 'Upon this rock,' he was pointing to himself." No, not according to the author's own logic. If Jesus were referring to Himself as the rock, He "should have" used a personal pronoun, and He "should have" said, "upon ME." Should He not?

Objection 4:
"Jesus is referred to as a petra elsewhere in the New Testament by none other than Petros himself (1 Peter 2:8) and also by Paul, the Apostle to the Gentiles (Romans 9:33; 1 Corinthians 10:4)."

Answer:
Which proves....what? Christ is referred to as a "stumbling-stone" and a "rock of scandal" sometimes, as we all know. So what? Why does that supposedly show that Peter cannot be the rock on which Christ built His Church? Besides, there is an unmistakable Greek word for "small stone," and that is lithos. If St. Matthew had intended to convey to his readers that St. Peter was only a small stone in contrast to "the rock," he certainly had an unmistakable word available to him. If we're going to emphasize the Greek over the Aramaic (fine with me; cf. Objection 6), then let's do it, but let's do it all the way! Christ would then have said, "You are Lithos ("stone"), and upon this petra ("rock") I will build my Church." Why didn't He?

Objection 5:
"It is actually more plausible, based on the context, that when Jesus said, "Upon this rock," he was pointing to himself specifically on his body that will lay the groundwork for the establishment of the Church later in Acts 2, that is, through his own sacrificial death, awesome resurrection and glorious ascension."

Answer:
No, that isn't more plausible at all. Especially the context shows the Catholic position to be correct. If Christ had referred to Himself, then why did He say "and upon this rock" and not "but upon this [or that] rock"? If Christ had wanted to indicate a contrast, then "but" would have been the conjunction to use to, not "and"! Why, in fact, did He change Simon's name at all, and why is Petros translated "rock" in Aramaic (Cephas, see St. John 1:42)?

Objection 6:
"The people in Galilee, where the conversation in Matthew 16:18 took place, spoke in Greek, it is therefore possible that their conversation happened in Greek."

Answer:
I guess it's possible that they spoke Greek, but Sacred Scripture indicates otherwise: "And taking the damsel by the hand, he saith to her: Talitha cumi, which is, being interpreted: Damsel (I say to thee) arise" (St. Mark 5:41). "Talitha cumi" is Aramaic, not Greek. Now, the interesting part is that Jesus was speaking to a family in Galilee (see St. Mark 5:20-21), so why didn't He speak in Greek, if the Protestant author is correct?

Anyway, in St. John 1:42, we have an Aramaic translation of Simon's new name, and it is Cephas, which means "rock." If Protestants were right and Peter was only a "little stone" (a petros), then the Aramaic word to use would have been "evna", not "cephas." Let's remember here that in Sacred Scriptures, name changes always occur when something very important takes place, for instance, when a man's role changes. Christ would not change Simon's name willy-nilly, especially not to Petros, a name that had never been given before!

Objection 7:
"Matthew often translates and expounds non-Greek terms of importance in his gospel, if Jesus meant Petros to be the foundational petra, Matthew should have translated and expounded it as he did in Matthew 1:23; 27:33, and 27:46."

Answer:
Why? There was no need for it! There were no Protestants around yet protesting anything! It was clear that Simon was the rock, the petra, but since he was male, the -a became an -os! Petros! Cephas! Peter! Rock!

Objection 8:
"The Holy Spirit has chosen specifically the ancient Greek language to accurately convey God's truth, thus, we should give weight on Greek evidence than on the Romanist Cephas-cephas evidence in Aramaic, an ancient translation of the Greek gospel from Matthew."

Answer:
The Holy Ghost hasalso chosen specifically the Aramaic language to reinforce Simon's new name: "And Jesus looking upon him, said: Thou art Simon the son of Jona. Thou shalt be called Cephas, which is interpreted Peter" (St. John 1:42). But granting that St. Matthew's Gospel was written originally in Greek, we do not need to assume it was written in Aramaic. We have all the evidence we need. Christ elicits a confession of faith from Simon, to which He responds that Simon is blessed because this was not revealed to him by any man but by the Father. Following this, he singles Simon out and tells him he now has a new name, a name which is based on the word "rock," but with a male ending since he is male, and that upon this rock (gee, which rock might that be??) He would build His Church, and that he (Simon Peter) would now receive the Keys to the Kingdom of Heaven! Come on, this is so crystal-clear, flowing so naturally, you have to want it to be false in order to argue against it. Which begs another question: do the Protestants concede that if St. Peter is the rock, then the papacy is true? Now that explains a lot, doesn't it....

The fact of the matter remains that even

Objection 9:
"If Jesus meant the Papacy to be an important aspect of Christianity, he should have made it as explicit as he made his return so explicit."

Answer:
Again, why? Isn't it explicit enough? It was taught by the Holy Apostles and by the Church Fathers! It was passed on and not seriously challenged until the ninth century (Photius), and even then, the papacy as such was not doubted but only the primacy of the papacy. But Protestants deny the Catholic hierarchy altogether (well, ok, depends on which Protestant you talk to)! Besides, does the Protestant really wish to go this route? We can easily use this argument and turn it against him about his perverse doctrine of "faith alone." Anyone remember St. James 2:24?

Objection 10:
"The Romanists use Matthew 16:18 to justify their Popery saying the Pope is the head of the Universal Church, this goes against the Scripture which says Jesus is the head of the Body (i.e., Universal Church) as in Colossians 1:18."

Answer:
You've got to be kidding! The Protestant author here could have simply picked up the Catechism of the Council of Trent to find the answer to this objection:

The Church has but one ruler and one governor, the invisible one, Christ, whom the eternal Father hath made head over all the Church, which is his body; the visible one, the Pope, who, as legitimate successor of Peter, the Prince of the Apostles, fills the Apostolic chair.

It is the unanimous teaching of the Fathers that this visible head is necessary to establish and preserve unity in the Church. This St. Jerome clearly perceived and as clearly expressed when, in his work against Jovinian, he wrote: One is elected that, by the appointment of a head, all occasion of schism may be removed. In his letter to Pope Damasus the same holy Doctor writes: Away with envy, let the ambition of Roman grandeur cease! I speak to the successor of the fisherman, and to the disciple of the cross. Following no chief but Christ, I am united in communion with your Holiness, that is, with the chair of Peter. I know that on that rock is built the Church. Whoever will eat the lamb outside this house is profane; whoever is not in the ark of Noah shall perish in the .flood.

The same doctrine was long before established by Saints Irenaeus and Cyprian. The latter, speaking of the unity of the Church observes: The Lord said to Peter, I say to thee, Peter! thou art Peter: and upon this rock I will build my Church. He builds His Church on one. And although after His Resurrection He gave equal power to all His Apostles, saying: As the Father hath sent me, I also send you, receive ye the Holy Ghost; yet to make unity more manifest, He decided by His own authority that it should be derived from one alone, etc.

Again, Optatus of Milevi says: You cannot be excused on the score of ignorance, knowing as you do that in the city of Rome the episcopal chair was first conferred on Peter, who occupied it as head of the Apostles; in order that in that one chair the unity of the Church might be preserved by all, and that the other Apostles might not claim each a chair for himself; so that now he who erects another in opposition to this single chair is a schismatic and a prevaricator.

Later on St. Basil wrote: Peter is made the foundation, because he says: Thou art Christ, the Son of the Living God; and hears in reply that he is a rock. But although a rock, he is not such a rock as Christ; for Christ is truly an immovable rock, but Peter, only by virtue of that rock. For Jesus bestows His dignities on others; He is a priest, and He makes priests; a rock, and He makes a rock; what belongs to Himself, He bestows on His servants.

Lastly, St. Ambrose says: Because he alone of all of them professed (Christ) he was placed above all.

Should anyone object that the Church is content with one Head and one Spouse, Jesus Christ, and requires no other, the answer is obvious. For as we deem Christ not only the author of all the Sacraments, but also their invisible minister ­­ He it is who baptises, He it is who absolves, although men are appointed by Him the external ministers of the Sacraments ­­ so has He placed over His Church, which He governs by His invisible Spirit, a man to be His vicar and the minister of His power. A visible Church requires a visible head; therefore the Saviour appointed Peter head and pastor of all the faithful, when He committed to his care the feeding of all His sheep, in such ample terms that He willed the very same power of ruling and governing the entire Church to descend to Peter's successors.

(See The Catechism of the Council of Trent, McHugh/Callan translation, TAN Books, 1982, pp.102-104)

The Pope is the visible head of the Church, who sits in the Chair of Peter until Christ returns. The parallels to Isaias 22 and the office of prime minister (in absence of the king) are striking! That Christ instituted a visible Church is obvious from His institution of the college of the Apostles, and the singling out of St. Peter specifically (cf. St. Matthew 18:15-18).

In his insistence that Peter cannot be the rock (because then the papacy would be established), the Protestant is asking us to accept the erroneous and absurd scenario of Christ first blessing and singling out Simon, saying that God the Father had revealed His Messiahship to him, then changing his name from Simon into "little stone," and then handing him the Keys to the Kingdom of Heaven!