Acts 13:48, KJV And when the Gentiles heard this, they were glad, and glorified the word of the Lord: and as many as were ordained to eternal life believed.
Acts 13:48, NKJV Now when the Gentiles heard this, they were glad and glorified the word of the Lord. And as many as had been appointed to eternal life believed.
Acts 13:48, ESV Now 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.
Acts 13:48, NASB And when the Gentiles heard this, they began rejoicing and glorifying the word of the Lord; and as many as had been appointed to eternal life believed.
I usually don't comment on prooftexts that have no other comments. However, the connection you are apparently seeking is not made in the verse. Although there is a positive correlation between believing and being appointed to eternal life according to Calvinist or non-Calvinist interpretation of the verses, the verses don't say they were chosen to believe.
Given the context, several other options are available.
1. Having accepted the good news for the Jews, the Gentiles believed it when it was offered to them. They were, therefore, already appointed to eternal life based upon their acceptance of Paul's teaching to the Jews. Note that Paul does not tell them the gospel
again. The only thing that happens is he says, "I now turn (offer it) to the Gentiles." This makes more sense of the fact that it mentions that they "rejoiced" and "honored the word of the Lord" (the equivalent of accepting the Good News in Luke-Acts) before it mentions that they had been appointed to eternal life. The Calvinist scenario would be better supported of the alternative order of wording had been provided: And as many as had been appointed to eternal life ---> rejoiced and honored the word of the Lord-----> and
then believed. But here is the actual order of what was said: "And the Gentiles rejoiced and honored the word of the Lord" ----> and as many as had been appointed to eternal life ----> believed. The connection you seek in the word order is not supported by the way Luke commented on the event.
2. Paul had just quoted the Old Testament, making an allusion to that very day, when God would bless the nations because of the disobedience of the Jews. Therefore, Luke sees God as intending to bless these very people when they believed, as opposed to simply blessing Jews. In other words, God knew about who would and wouldn't believe and spoke of the day when he would give eternal life to these very people, thereby fulfilling God's promise to do something that they "would never believe, even if someone had told you" (v. 41). The Jews' inheritance (i.e., eternal life - associated by the Jews with the coming kingdom) is given to these very Gentiles when they believe, just as God had planned, and to the disbelief of the Jews in attendance.
3. Middle voice interpretation of
tasso being "as many has had
ranged themselves in order for eternal life believed." Or a passive voice, with the same intent, reflecting upon the culmination of what had just happened and the resulting affect it had on some of the listeners.
The first two options make the most sense to me. But there are a few others I would provide if I had the time.