One Protestant argument I have encountered holds that because Scripture elsewhere calls Jesus Christ a “Rock” — therefore (so the reasoning goes) Peter can’t possibly be called the same thing. 

This simply isn’t true. The objection presupposes what I would argue is an unbiblical and hyper-rationalistic “either/or” outlook, whereas the Bible teaches a “both/and” point of view. Here are some of the passages brought up in order to set forth such a view:

• Matthew 21:42 Jesus said to them, “Have you never read in the Scriptures: ‘The very stone which the builders rejected has become the head of the corner; this was the Lord’s doing, and it is marvelous in our eyes?’” (cf. Mark 12:10; Luke 20:17-18)
• Acts 4:11 As it is written, “Behold, I am laying in Zion a stone that will make men stumble, a rock that will make them fall; and he who believes in him will not be put to shame.”
• Romans 9:32-33 Why? Because they did not pursue it through faith, but as if it were based on works. They have stumbled over the stumbling stone, as it is written, “Behold, I am laying in Zion a stone that will make men stumble, a rock that will make them fall; and he who believes in him will not be put to shame.”
• 1 Corinthians 10:4 ... For they drank from the supernatural Rock which followed them, and the Rock was Christ.
• 1 Peter 2:4, 6-8 Come to him, to that living stone, rejected by men but in God’s sight chosen and precious; ... For it stands in scripture: “Behold, I am laying in Zion a stone, a cornerstone chosen and precious, and he who believes in him will not be put to shame.” To you therefore who believe, he is precious, but for those who do not believe, “The very stone which the builders rejected has become the head of the corner,” and “A stone that will make men stumble, a rock that will make them fall”; for they stumble because they disobey the word, as they were destined to do.

Now, is all of this this intended to exclude anyone else being called a rock or a stone? No. In Scripture, creatures are often called in an essentially lesser sense or degree, things which God is called. There can be such a thing as a Big Rock (God) and a small rock or stone (men or a man). In fact, this is explicit biblical teaching. Note that in the passage from 1 Peter, above, I left out a verse. Here it is:

• 1 Peter 2:5 And like living stones be yourselves built into a spiritual house, to be a holy priesthood, to offer spiritual sacrifices acceptable to God through Jesus Christ.

The “spiritual house” is likely referring to the Church. Jesus, in the larger passage, was called a “living stone” a “cornerstone” and “the head of the corner.” Yet we Christians are also called “living stones” in the same passage. Thus, there is no “either/or” pattern here. Both things can be true. If there can be little stones along with God as the Big Rock, then there can also conceivably be the “chief” of these secondary stones, and that would be Peter, based on the data of Matthew 16:18. And there’s more, too:

• Ephesians 2:19-22 So then you are no longer strangers and sojourners, but you are fellow citizens with the saints and members of the household of God, built upon the foundation of the apostles and prophets, Christ Jesus himself being the cornerstone, in whom the whole structure is joined together and grows into a holy temple in the Lord; in whom you also are built into it for a dwelling place of God in the Spirit.

This is again referring to the Church (2:19), which is “built upon the foundation of the apostles and prophets” (2:20). Thus we have precisely the same notion that Catholics contend is expressed in Matthew 16:18: there is a secondary / “co-worker” sense in which the Church is built upon men. Here it is the twelve apostles and prophets as well. In Matthew it’s Peter, as the leader and foremost of the apostles. And then in the next verse we have the “Big Rock” — “Christ Jesus himself being the cornerstone.” So there is no false dichotomy. One is not in opposition to the other. Another passage teaches the same thing (without mentioning Jesus in this particular instance):

• Revelation 21:14 And the wall of the city had twelve foundations, and on them the twelve names of the twelve apostles of the Lamb.

This sort of typology is common in Scripture: 

• Jesus is the Creator, but we participate in procreation: helping to cause human beings to begin existing through sexuality. 
• Jesus is the Shepherd of the Church (John 10:16), yet some Christians leaders are called shepherds, too (John 21:15-17; Eph 4:11; 1 Peter 5:4).
• Jesus forgives and redeems us and reconciles us to God the Father (2 Corinthians 5:18-21), but we also share in offering forgiveness and reconciliation (Matthew 9:5-8; 18:18; John 20:21-2; Acts 2:38; 2 Corinthians 5:18-20; James 5:14-15).
• We are “co-workers” with God (“we are God’s fellow workers”: 1 Corinthians 3:9; “Working together with him”: 2 Corinthians 6:1; ” the Lord worked with them”: Mark 16:20; “I worked harder than any of them, though it was not I, but the grace of God which is with me”: 1 Corinthians 15:10; “work out your own salvation with fear and trembling; for God is at work in you”: Phil 2:12-13).
• God even shares his glory with us (John 17:22; Romans 5:2; 9:23; 2 Corinthians 3:18; 1 Thessalonians 2:12; 2 Thessalonians 2:14; 1 Peter 4:14; 5:1; 2 Peter 1:3-4)