You have that exactly backwards according to scripture:
Act 13:48 And when the Gentiles heard this, they began rejoicing and glorifying the word of the Lord, and as many as were appointed to eternal life believed.
They were not appointed to eternal life BECAUSE they believed. They believed because they were appointed to eternal life. Otherwise Acts is wrong.
No, just a statement of foreknown fact.
Act 13:48 And when the Gentiles heard this, they began rejoicing and glorifying the word of the Lord, and as many as were appointed to eternal life believed.
They were not appointed to eternal life BECAUSE they believed. They believed because they were appointed to eternal life. Otherwise Acts is wrong.
In context, Acts 13 represents the first public preaching to the Gentiles. From verses 16-45, Paul and Barnabas preach to the Jews, and NOT the Gentiles. But when they reject the Gospel, we find verses 46-48:
46And Paul and Barnabas spoke out boldly, saying, “It was necessary that the word of God be spoken first to you. Since you thrust it aside and judge yourselves unworthy of eternal life, behold, we are turning to the Gentiles. 47 For so the Lord has commanded us, saying, “’I have made you a light for the Gentiles, that you may bring salvation to the ends of the earth.’” 48And when the Gentiles heard this, they began rejoicing and glorifying the word of the Lord, and as many as were appointed to eternal life believed.
A POSSIBLE interpretation is the one HarleyD advances - that they were ordained to salvation as individuals, and then believed. And if this was a common phrasing of the conversion experience, that interpretation would have a lot of strength.
However, another POSSIBLE interpretation notices that 30 verses were addressed to the Jews, and only in verse 46 does it change. And what did P&B say to the Jews? That they “judge [themselves] unworthy of eternal life” - which doesn’t sound like irresistible grace.
In verse 47, they proclaim that they are also sent to the Gentiles, and in 48 “when the Gentiles heard this, they began rejoicing and glorifying the word of the Lord, and as many as were appointed to eternal life believed.”
So another interpretation is that ‘as many as were appointed’ refers to the fact that it wasn’t just Jews receiving the message, but the Gentiles who were also appointed for salvation by God. Until this point, only Cornelius has received the Gospel as a Gentile, and he was a special case: “a devout man who feared God with all his household, gave alms generously to the people, and prayed continually to God.”
There is at least one other way of looking at it, found at this link:
http://classicalarminianism.blogspot.com/search/label/Exegeting%20Acts%2013%3A48