There are several indications that Peter was not the leader. First, he is not mentioned in Acts after Acts 15:7, while James is obviously in authority in Acts 21:18. When Peter was freed from prison, he told those who had been praying for his release, to go tell James he had be set free by an angel (Acts 12:7-17). Another clue is that James, the half-brother of Jesus, was the one who made the final decision in the debate over circumcising the Gentile believers (Acts 15:19). The word "sentence" is translated "judge or judgment" eighty-eight times, and "sentence" only three times (Acts 15:19; Lk. 23:24; 2 Cor. 1:9), all of which indicate the final say in a matter. And then, there is the order of names: James is named first when Paul and "his company" reported in Jerusalem (Acts 21:8, 18), and also listed first in Paul's sarcastic statement about James, Peter, and John being "pillars" of the Church (Gal. 2:9).
And, Peter frequently demonstrated his lack of spirituality. He was called "Satan" by Jesus after he had been told he would control who was accepted in God's kingdom (Mt. 16:23). He attacked and cut off the ear of the high priest's servant, one minute, and then he denied he knew Jesus three times (Jn. 18:10, 18-27). And finally, there is Peter's hypocrisy when a group of men, sent by James, arrived in Antioch (Gal. 2:11-14).
Since a Pope is said to infallible and to answer only to God,
James would be much more likely to be the "first pope," than would Peter.