John 21.15-22

15 ¶ So when they had dined, Jesus saith to Simon Peter, Simon, son of Jonas, lovest (αγαπας, agapas) thou me more than these? He saith to him, Yea, Lord; thou knowest that I love (φιλω, filo) thee. He saith to him, Feed my lambs.
16 He saith to him again the second time, Simon, son of Jonas, lovest (αγαπας, agapas) thou me? He saith to him, Yea, Lord; thou knowest that I love (φιλω, filo) thee. He saith to him, Feed my sheep.
17 He saith to him the third time, Simon, son of Jonas, lovest (φιλεις, fileis) thou me? Peter was grieved because he said to him the third time, Lovest (φιλεις, fileis) thou me? And he said to him, Lord, thou knowest all things; thou knowest that I love (φιλω, filo) thee. Jesus saith to him, Feed my sheep.
18 Verily, verily, I say to thee, When thou wast young, thou didst gird thyself, and didst walk where thou wouldest: but when thou shalt be old, thou shalt stretch forth thy hands, and another shall gird thee, and carry thee where thou wouldest not.
19 This he spoke, signifying by what death he should glorify God. And when he had spoken this, he saith to him, Follow me.
20 ¶ Then Peter, turning about, seeth the disciple whom Jesus loved following; who also leaned on his breast at supper, and said, Lord, who is he that betrayeth thee?
21 Peter seeing him saith to Jesus, Lord, and what shall this man do?
22 Jesus saith to him, If I will that he shall tarry till I come, what is that to thee? follow thou me.

Rome's interpretation of this passage of Scripture:

And it was to Peter alone that Jesus, after his resurrection, confided the jurisdiction of supreme pastor and ruler of his whole fold, saying: Feed my lambs, feed my sheep.
Vatican I, First Dogmatic Constitution on the Church of Christ; Chapter 1

In this Church of Christ the Roman Pontiff, as the successor of Peter, to whom Christ entrusted the care of his sheep and his lambs, has been granted by God supreme, full, immediate and universal power in the care of souls.
Vatican II, Decree on the Bishops' Pastoral Office in the Church

Commentary:

This passage of Scripture is another of the pillars upon which Rome attempts to build her claim to supremacy and superiority over all the Body of Christ. Once again, their claims cannot hold up under scrutiny. Rome's interpretations above read far more into this passage than is written, and far more than can reasonably be deduced from it. But how readily do her many adherents accept and embrace those interpretations as though they were divine truth.

Let's take a look at the passage in its context.

Before the Crucifixion, Peter confidently boasted to Jesus that, "though all men shall be offended because of thee, yet will I never be offended" (Matthew 26.33). He thus professed that his love for Christ was greater than others' love. To this Jesus responded by telling Peter that he would deny him three times before morning, which came to pass. This was a serious failure on Peter's part. Christ had taught them that whoever would deny him before men, he would deny that one before his Father in heaven (Matthew 10.33). Thus, after the Resurrection, when Christ asked Peter three times if he loved him, he was but restoring him after his thrice denial. There simply is no way to read into this passage that Jesus was making Peter the supreme shepherd over the whole Church—much less that the words could apply to the Roman bishop today.

When Jesus questioned Peter three times, the first two times he used the word agape, which means a deeper, self-sacrificing, love (as the love of Christ that took Him to Calvary); and, in the first question, he used the words "more than these" (verse 15), alluding to Peter's confident boast (Matthew 26.33). To both of these first two questions Peter responded, not that he had agape, but that he had fileo (filial love), which is a lesser depth of love. The third time Jesus asked Peter if he had but fileo for Jesus, which question grieved Peter. To this Peter responded again that, yes, he had fileo for Jesus. This was to show Peter that his love for Christ was not as great as he had professed it to be. Jesus walked Peter down the ladder from agape love and "more than these", which sort of love he had first professed to have, to fileo love, which was where Peter's love actually was. This cannot be seen in the English translation, but see the passage above. At any rate, rather than making Peter a supreme singular shepherd over the entire Body of Christ, Jesus was but restoring him after his thrice denial. Christ alone, who could say, "I am the Good Shepherd" (John 10.11), is the supreme Pastor over his Church.

105. ... For which reason, also, Peter in the Gospel is asked three times whether he loves the Lord, that by the threefold answer the bonds of the guilt he had contracted by denying the Lord might be loosed....
Ambrose of Milan 339/340 - 397, On the Holy Spirit, Book II, Chapter X, § 105

2. It is Peter … who merits thrice to hear the words: "Feed My little lambs; feed My lambs; feed My sheep." And so, by feeding well the flock of Christ with the food of faith, he effaced the sin of his former fall. For this reason is he thrice admonished to feed the flock; thrice is he asked whether he loves the Lord, in order that he may thrice confess Him, Whom he had thrice denied before His Crucifixion.
Ambrose of Milan 339/340 - 397, Exposition of the Christian Faith, Book V, Prologue, § 2

… Therefore, after His resurrection the Lord questioned him, not as being ignorant with what a heart he would confess the love of Christ, but that he might by a threefold confession of love, efface the threefold denial of fear.
Augustine 354 - 430, Sermons on Selected Lessons of the New Testament, Sermon LXXXVII, § 3

Further, there is more evidence in this passage that Jesus was not making Peter a supreme pastor over the entire Church. When Peter began to inquire about John (which he would have reason to do if he were to be his supreme pastor), Jesus told him in essence that it was not his concern what John was to do (verse 22); but Peter's responsibility was to follow Christ. Rather than placing the entire Church, including John, under Peter's pastoral care, Jesus was telling Peter to take heed to his own walk with the Lord, and let the Lord watch out for John.

God wills that, in Christ, the truth that sets us free might be made known, and that we might draw near to God worshipping freely, and joyfully, in Spirit and in truth. Let us do so with rejoicing.

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