The ROCK is not meant for Peter, it was Jesus.
Matt 21:44 And whosoever shall fall on this stone shall be broken: but on whomsoever it shall fall, it will grind him to powder.
Jesus Christ "the chief cornerstone" (Matthew 21:42). Jesus was talking about building His church upon the solid bedrock, not a small pebble.
"Therefore whosoever heareth these sayings of mine, and doeth them, I will liken him unto a wise man, which built his house upon a rock: And the rain descended, and the floods came, and the winds blew, and beat upon that house; and it fell not: for it was founded upon a rock." (Matthew 7:24-25)
Paul is clear in 1 Corinthians 3:11 "For no other foundation can anyone lay than that which is laid, which is Jesus Christ." We can't look to a mere human being as the foundation of the Christian church! "It is better to trust in the Lord than to put confidence in man." (Psalm 118:

. Like David said, "The Lord is my rock" (Psalm 18:2) "And who is a rock, except our God?" (Psalm 18:31).
Peter was not "rock solid" at this point in time. If we read on in Matthew 16, just a few verses after Jesus spoke of the rock, we find Christ rebuking Peter for trying to hinder His ministry: "But He turned and said to Peter, Get behind Me, Satan! You are an offense to Me, for you are not mindful of the things of God, but the things of men." (Matthew 16:23) Later, it was Peter who denied three times that he even knew Jesus (Matthew 26:69-75). So he obviously had some growing to do in his relationship with the Lord, and Jesus knew that (Luke 22:31-32).
Now it is true that every believer of Jesus Christ is a stone in the great building of God, His church. Peter himself tells us that Jesus is the "cornerstone", and we are also stones resting on the foundation of Christ: "Coming to Him as to a living stone, rejected indeed by men, but chosen by God and precious, you also, as living stones, are being built up a spiritual house, a holy priesthood, to offer up spiritual sacrifices acceptable to God through Jesus Christ. Therefore it is also contained in the Scripture, Behold, I lay in Zion A chief cornerstone, elect, precious, And he who believes on Him will by no means be put to shame." (1 Peter 1:4-6)
Successor to Peter
church (chûrch) KEY
NOUN:
A building for public, especially Christian worship.
often Church
The company of all Christians regarded as a spiritual body.
A specified Christian denomination: the Presbyterian Church.
A congregation.
Public divine worship in a church; a religious service: goes to church at Christmas and Easter.
The clerical profession; clergy.
Ecclesiastical power as distinguished from the secular: the separation of church and state.
TRANSITIVE VERB:
churched, church·ing, church·es
To conduct a church service for, especially to perform a religious service for (a woman after childbirth).
ADJECTIVE:
Of or relating to the church; ecclesiastical.