Jesus did not appoint Peter to the headship of the apostles and forbade any such notion. (Read Luke 22:24-26; Ephesians 1:22-23; Colossians 1:18; 1st Corinthians 3:11). There is no mention in Scripture or history that Peter ever was in Rome, much less that he was pope there for 25 years; Clement, 3rd bishop of Rome, remarks that "there is no real 1st century evidence that Peter ever was in Rome."
We have all heard it. The denial that St. Peter was made the Vicar of Jesus Christ. The denial that Christ gave to him the Headship over the Apostles. This denial, of course, is needed in order to legitimise the falsely named "Reformation". Because, if Peter is truly the head, then to break from that head would prove one to be outside the Church, and thus outside the saving Faith of Jesus Christ. But, it is common sense that there should be an earthly head.
Whilst on earth, Our Lord was the visible sign of unity, around Whom the Apostles and other disciples gathered, sitting at the feet of that great Teacher and Master of souls. Around the feet of the Divine Preacher, the whole Church gathered, and still gathers, as if lying under the shade of a tree, listening the gentle Voice of Her Spouse. Whilst on earth, He was visible, and therefore the visible Head on earth. But, what should happen after the Ascension, when Christ would ascend to the Father, and be seen no more? The natural question would be: who did Jesus leave in charge on earth?
Think of it like an Empire. The Emperor appoints governors for his Empire, and endows them with his authority. In the same way, Christ bestowed authority on one Apostle in particular, not to supersede Him, but to rule in His Name. This Apostle was St. Peter, who the Holy Church calls "the Prince of the Apostles". The Successor of St. Peter is the Pope, the Supreme Sovereign Pontiff.
So, as around Christ, the whole family of the Church is gathered around the Prince of the Apostles, listening to him, because through him, Christ speaks. When anyone in authority departs from his usual place, he always leaves someone in charge. So, too, did Christ likewise. Specially more so with the Church, since She is the only way to Heaven. What great confusion would there be if Christ left no one in charge, and each Apostle and disciple argued amongst themselves about whom was to take charge, finally leaving one another to found their own churches! What scandal would that be to the Unity of Christ? And what manifest contradictions would there be in what doctrines are to be believed?! If that sounds like Protestantism, well, you now know the cause of the illness! For this reason did He leave St. Peter to be Head of the Church on earth, not to replace Christ, but to speak in His place and to bind the faithful to that saving Truth which Christ entrusted to His Catholic Church (called the Deposit of Faith).
It would be absurd to think, as the Protestants seem to, that Christ left the Church without a governor, without someone we could turn to for guidance and instruction! Is Christ so without heart so as to abandon His beloved Church to be grappled and torn asunder by men? Is He with love that He would leave His sheep without a visible shepherd to follow? Would He leave us in uncertainty about doctrine, so that we wouldn't know the correct way to believe in Him? This is seemingly the Protestant Christ, so heartless, cold and cruel, as to leave poor children without any help. But the true Christ, the Catholic Christ, as it were, is not so heartless, but ever full of compassion gave to us our Shepherd, the man whom we call our Holy Father - the Pope!
I shall use herein the Douay-Rheims Catholic Bible in the main for the Scriptural text. Any other version shall be duly noted. Verses from the DR shall be linked to for your reading.
(1) JESUS & PETER: