The conversion of Cornelius only comes after yet another vision given to Simon Peter (Acts 10:10-16) himself.

In the vision, Simon Peter sees all manner of four-footed beasts and birds of the air being lowered from Heaven in a sheet.

A voice commands Simon Peter to eat.

When he objects to eating those animals that are unclean to Mosaic Law, the voice tells him not to call unclean that which God has cleansed or purified.

When Cornelius' men arrive, Simon Peter understands that the vision permits the conversion of the Gentiles.

When Cornelius himself meets Simon Peter, Cornelius falls at his feet in adoration.

Simon Peter, picking Cornelius up, welcomes him.

After the two men share their visions, and Simon Peter tells of Jesus' ministry and the Resurrection, the Holy Spirit falls on everyone at the gathering.

The Jews among the group (presumably they were all Jews if Cornelius was the first gentile convert) are amazed that Cornelius and other uncircumcised should begin speaking in tongues, praising God.

Simon Peter thereupon orders that Cornelius and his followers be baptized.

The controversial aspect of Gentile conversion will taken up later at the Council of Jerusalem (Acts 15), but has its roots in the concept of "proselytes" in the Septuagint (the Greek translation of the Hebrew Bible) and Jewish Noahide Law.