One presumes that Cornelius, his relatives, and friends are believing the word as it is spoken but Luke simply says, “the Holy Spirit fell on all who heard the word”.
Peter and the believers who were with him “were amazed, because the gift of the Holy Spirit was poured out even on the Gentiles”. They had no idea that God would do this. How did they know that the Spirit had been given to the Gentiles? Because the Gentiles were speaking in tongues and extolling God – the same as they had done at Pentecost. Peter, putting together, the vision he had seen, the Spirit’s command to go, and the obvious pouring out of the Spirit – recognizes that these Gentiles had been accepted by God and therefore baptizes them in the name of Jesus and remains some days with them teaching them. In other words, Gentiles are now in union with Jesus and have become part of his people.
An aside: On this occasion, as on the Day of Pentecost, the pouring out of the Holy Spirit results in tongue speaking – although it also results in proclaiming God’s work. Some have argued from this that whenever a person receives the Holy Spirit they should speak in tongues. Counter to this is that there is no record of the Samaritans speaking in tongues when they received the Spirit. Also, the context here suggests another reason why Cornelius and the Gentiles speak in tongues. Samaritans were like half-Jews so while their conversion is a surprise it is not beyond the realms of the imaginable. The conversion of Gentiles on mass who then remain as Gentiles was unheard of. The Holy Spirit’s being poured out on the Gentiles and their speaking in tongues is to identify them as having received the same experience as the Jewish believers at Pentecost. Thus, there is a special reason why in this case the Gentiles speak in tongues and it is not necessarily the usual way things occur.
Supporting the above we find in chapter 11 that when the believers in Jerusalem here about Peter’s fellowshipping with these new believers they need convincing. Peter then recounts his vision and Cornelius’ vision – this is the third time this story is told – once when it happened, once when Peter and Cornelius recount what they had experienced to one another, and now to the Jerusalem church. The threefold repetition is to indicate that it is important. The climax of Peter’s argument though is the pouring out of the Holy Spirit as a sign of God’s acceptance: “If then God gave the same gift to them as he gave to us when we believed in the Lord Jesus Christ, who was I that I could stand in God’s way?”
This brings all disputing to an end: “When they heard these things, they fell silent. And they glorified God, saying, “Then to the Gentiles also God has granted repentance that leads to life.”