“If then God gave to them the like gift as he did also to us, when we believed on the Lord Jesus Christ, who was I, that I could withstand God?”
Thus, he asked, what should he have done? If God gave to the Gentiles the same gift as He had given at Pentecost to all who believed on the Lord Jesus Christ, who was he to withstand God? God had thereby made it clear that He had cleansed these unclean Gentiles so that they were acceptable to Him. Thus they were no longer common or unclean. They were precious to God and acceptable to Him, and they were that just as they were, in their uncircumcised state. And just as they were they were a part of God's holy nation.
So Peter has made clear that the initiative was God's each step of the way. It was God Who had put him into a trance and given him his vision. It was His Spirit Who had bid him go to Cornelius. It was the Angel who had told Cornelius what to do. The coming of the Holy Spirit on them had been as a result of God's direct and unexpected action. Nothing therefore had been of Peter's doing.
It should be noted that it was not Peter's authority that was being accepted here, it was his logic combined with the facts. Thus the other Apostles were willing in the end to put their authority behind his actions. They too knew what it meant for God to direct them, so that not to do so would have been to go against God.