Is the Pope really Christ's Representative on earth?

I have often been asked if the Pope is really the representative of Christ on earth. The only passage I know that the Roman Catholics take this idea from is found in Matthew 16. I will add the whole passage below: Matt 16:13When Jesus came to the region of Caesarea Philippi, he asked his disciples, "Who do people say the Son of Man is?" 14They replied, "Some say John the Baptist; others say Elijah; and still others, Jeremiah or one of the prophets." 15"But what about you?" he asked. "Who do you say I am?" 16Simon Peter answered, "You are the Christ,[b] the Son of the living God." 17Jesus replied, "Blessed are you, Simon son of Jonah, for this was not revealed to you by man, but by my Father in heaven. 18And I tell you that you are Peter,[c] and on this rock I will build my church, and the gates of Hades[d] will not overcome it.[e] 19I will give you the keys of the kingdom of heaven; whatever you bind on earth will be[f] bound in heaven, and whatever you loose on earth will be[g] loosed in heaven." 20Then he warned his disciples not to tell anyone that he was the Christ. Roman Catholics argue that this means Peter is the first Pope. Protestants argue this simply means the Church will be built upon the revelation of who Jesus is by the Father in heaven.