Paul publicly rebuked Peter (Galatians 2:1114). This controversy in Antioch led directly to the Council of Jerusalem (Acts 15:129).
As for the book of Matthew, let us turn to the Fathers...
Blessed Theophylact: "The Lord is saying, 'This confession which you have made shall be the foundation of those who believe.'" [P.G. 123:85B (col. 320).]
Saint Chrysostom: "'Upon this rock I will build'; that is, on the faith of the confession." [Hom. 54, P.G. 58:518 (col.534).]
Saint Leo: "Upon this firmness, He says, I shall raise my Temple, and it will rise upon the steadfastness of this faith, and the loftiness of My Church will mingle with the heavens. The gates of Hades shall not master this profession; nor the bonds of death bind it. For these words are the words of life, and as they raise those who confess them up to heaven, so they plunge those that deny them down to hell." [Sermon 83(2), P.L. 54 (col. 429), in FC, 93:357; Toal, III:267, 268.]
"This one (Peter) is called a rock in order that on his FAITH (Rock) he may receive the foundations of the Church." - St. Gregory Nazianzen, 26th Discourse
"The Rock on which Christ will build His Church means the faith of confession." - St. John Chrysostom, 53rd Homily on St. Matthew
"The Rock (petra) is the blessed and only rock of the faith confessed by the mouth of Peter. It is on this Rock of the confession of faith that the Church is built." - St. Hilary of Poitiers, 2nd book on the Trinity
Hilary wrote the first lengthy study of the doctrine of the Church in Latin. Proclaimed a "Doctor of the Church" by the Roman See in 1851, he is called the Athanasius of the Western Church.
'"The word "Rock" has only a denominative value-it signifies nothing but the steadfast and firm faith of the apostles."
- Cyril of Alexandria, Upon St.John, Book JJ, Chap. XII
In his Letter to Nestorius, St. Cyril says: "Peter and John were equal in dignity and honor. Christ is the foundation of all -the unshakeable Rock upon which we are all built as a spiritual edifice."
"Faith is the foundation of the Church, for it was not of the person but the faith of St. Peter of which it was said, 'the gates of hell shall not prevail'; certainly it is the confession of faith which has vanquished the powers of hell."
"Jesus Christ is the Rock. He did not deny the grace of His name... to Peter because he borrowed from the Rock the constancy and solidity of his faith- thy Rock is thy faith, and faith is the foundation of the Church. If thou art a Rock, thou shalt be in the Church, for the Church is built upon the Rock... (the profession of faith in Christ Jesus)." - St. Ambrose: The Incarnation