In Matthew 16:18, Jesus said to Simon, “I also say to you that you are Peter, and upon this rock I will build My church; and the gates of Hades will not overpower it.”
Roman Catholics interpret Matt. 16:18 to mean that Peter is the rock upon which the church is built. That interpretation then becomes the basis for the doctrine of papal succession. If Peter is the rock on which the church is built, and if the bishops of Rome are Peter’s successors, then it follows, they say, that the papacy remains the foundation of the church.
But that is not at all what Matthew 16:18 teaches.
The name “Peter” was a nickname given to Simon by Jesus, all the way back in John 1:42 when Peter first met Jesus. Coming from the Greek word petros (or the Aramaic word “Cephas”), the name Peter means “Rock” or “Stone.” To use an English equivalent, Peter means “Rocky.”
But when Jesus said, “I say to you that you are Peter, and upon this rock I will build My church,” He differentiated between Peter and the “rock” by using two different Greek words. The name Peter is petros, but the word for “rock” is petra.
Those terms may sound similar to us, but ancient Greek literature shows that they actually refer to two different things. Petros was used to signify a small stone; petra, by contrast, referred to bedrock or a large foundation boulder (cf. Matt. 7:24-25).
So, to paraphrase Jesus’ words, the Lord told Peter, “I say to you that you are a small stone, and upon this bedrock I will build My church.” It was a play on words that made a significant spiritual point.
What then was the bedrock to which Jesus was referring? The answer to that question comes a couple verses earlier in Matthew 16.
Matthew 16:13–17: Now when Jesus came into the district of Caesarea Philippi, He was asking His disciples, “Who do people say that the Son of Man is?”  And they said, “Some say John the Baptist; and others, Elijah; but still others, Jeremiah, or one of the prophets.”  He said to them, “But who do you say that I am?”  Simon Peter answered, “You are the Christ, the Son of the living God.”  And Jesus said to him, “Blessed are you, Simon Barjona, because flesh and blood did not reveal this to you, but My Father who is in heaven.”
Peter was just a small stone built atop the bedrock of something much bigger than himself: namely, the truth that Jesus is the Christ the Son of the living God. Put simply, Peter was not the rock; Christ is the Rock. And as Peter and the other apostles testified to the truth about Christ (which Peter did in verse 16), the church was built upon its only sure foundation.
The rest of the New Testament bears this out.
In 1 Corinthians 3:11, Paul wrote that “no man can lay a foundation other than the one which is laid, which is Jesus Christ.”
In Ephesians 2:20, Paul further explained that Jesus Christ is the cornerstone on which the church is founded by the apostles.
Even Peter himself, in 1 Peter 2:1–10 compared all believers to small stones that are part of the superstructure of the church. By contrast, Peter noted in vv. 6, 7, the Lord Jesus is the cornerstone on which the church is built. Peter said the same thing to the Jewish religious leaders in Acts 4:11. Speaking of Jesus, Peter proclaimed, “He is the stone which was rejected by you, the builders, but which became the chief corner stone.”
If we were to go beyond Peter’s lifetime, and consider the writings of the church fathers from Origen to Chrysostom to Augustine – we would likewise find that the vast majority of ancient interpreters did not view the rock in Matthew 16:18 as a reference to Peter. The church fathers generally understood the “rock” to refer either to the apostles collectively, or to the specific content of Peter’s confession. In either case, they understood that Matthew 16:18 ultimately centered on Christ – the One to whom the apostles testified, and the One to whom Peter’s confession pointed.
Thus, we see the Roman Catholic understanding of Matthew 16:18 falls short on at least four levels:
1) Grammatically, it does not account for the lexical distinction between petros (Peter) and petra (rock).
2) Contextually, it makes Peter the focal point of Matthew 16, when the text is clearly featuring truth about Jesus.
3) Theologically, it tries to make Peter the rock when the rest of the New Testament declares Christ to be the Rock.
4) Historically, the Roman Catholic view is not the patristic view of the first few centuries.
(Moreover, even if Peter were the "rock" of Matthew 16:18, such an interpretation would still not necessitate the notion of papal succession. But that is the topic of another post.)
Peter’s nickname might have been Rocky, but Peter himself understood that the Rock was Jesus Christ. The Rock on which Peter’s life was built was none other than the Rock of Salvation; the Rock of Deliverance; the Chief Cornerstone; and the Rock of Ages.
Peter bore witness to that truth in Matthew 16:16. The rest of the Apostles bore witness to that throughout their ministries. And it was the truth of that apostolic witness to Jesus Christ that formed the foundation of the church.