In the first reading (Acts 10:25-26, 34-35, 44-48), we see Peter coming into his role as the earthly head of the church - the Pope. His words to Cornelius are used as an argument against the papacy.
There are two problems with that approach to the passage. First, Peter is establishing something the is true about the Pope - that he is not a king or god to be worshipped. He is not one to be served but one to serve. He is the beautifully paradoxical "first among equals" with the other bishops, the successors to the apostles.
Second, those that cite this passage as evidence that Peter is denying his papal authority overlook what he does next; he makes an authoritative statement. He declares that all people - the Gentiles, not just the Jews - receive baptism. He ordered it.