While Peter was still saying this, the Holy Spirit fell on all who heard the word. And the believers from among the circumcised who came with Peter were amazed, because now the gift of the Holy Spirit was even being poured out on Gentiles. For they heard them speaking in tongues and extolling God. Then Peter declared, "Can any one forbid water for baptizing these people who have received the Holy Spirit just as we have?" And he commanded them to be baptised in the name of Jesus Christ. Then they asked him to remain for some days.
All through this story Luke tells us how the Holy Spirit was actively leading the way. But why does He work in peculiar ways like with Peter's dreams and the angel's visit. Somehow God does a mighty work in transforming our relationships but He doesn’t reveal Himself in a way that over-rides the customs which form and shape the relationships we have with one another. They are still our responsibility. The coming of the Holy Spirit into the personal and corporate lives of those who believe in Jesus unleashes a new power by which such customs can be transformed and become a means of His blessing. Recall that Jesus didn't "gate-crash" the Samaritans on his way up to Jerusalem. Their customs may have prevented them from showing Him hospitality. But, in the fullness of time, God's Spirit accompanied Philip's work and those communities were transformed. People in that region came to realise that their Messiah had come - even if they hadn't put out the welcome mat for Him when He walked by.
In Caesarea, the Holy Spirit did not wait for Peter to finish his explanation, as if this was a liturgy in which God would play His part after Peter finished his lines. That is not how it is recorded. As Peter came to the end of what he had to say, Peter's Jewish companions became aware that they were able to hear what these Gentiles were saying in a new way. This was a similar experience to what had amazed them on the day of Pentecost when they had heard God's word spoken in their own tongues. Clearly, this was the same Spirit who had made Himself known to them then.
It was not only important for Gentile believers to receive the Holy Spirit, also it was important that believing Jews witness the outpouring of God's Spirit on these non-Jewish believers in Jesus Christ. The Son of Man was a Jew. Salvation was from the Jews. But God had made this salvation available to all who believe. And so, the Holy Spirit poured out on this gathering, leaving Peter's companions amazed. So he said: baptise these people. They have received the Holy Spirit too, just like we did, so let us proceed to baptise them into the name. They are equally members of Jesus Christ, just as we are. The coming of the Lamb of God who takes away the sin of the world is not done in some corner. This event confirms and demonstrates in a new way God's long-term purposes for His creation. It confirms the place His Son has for us in this Kingdom. And so, up and down the Mediterranean coast, believers witnessed a mighty work of God. Gentiles had received the gift of the Holy Spirit.