Some of you folks understand the Greek. That is fine; I wish I did. I can use a lexicon to find how a Greek or Hebrew word is translated in the Bible but that is the extent of my linguistic ability as far as Greek is concerned. What I can do is compare various translations. But! I must then trust that the translators are both competent and honest! I use the KJV primarily but I also checked Acts 13:48 in a number of other translations [and one concise paraphrase].
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.
ASV: And as the Gentiles heard this, they were glad, and glorified the word of God: and as many as were ordained to eternal life believed.
GLT: And hearing, the nations rejoiced and glorified the Word of the Lord. And as many as had been appointed to eternal life believed.
KJ21: 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.
MKJV: And hearing, the nations rejoiced and glorified the Word of the Lord. And as many as were ordained to eternal life believed.
NASB95: 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.
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.
NIV: When the Gentiles heard this, they were glad and honoured the word of the Lord; and all who were appointed for eternal life believed.
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.
YLT: And the nations hearing were glad, and were glorifying the word of the Lord, and did believe--as many as were appointed to life age-during;
ESV: 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.
Berk: When they heard this, the Gentiles were glad; they eulogized the Lord’s message and believed - as many as were appointed for eternal life.
There are others I could check but those were sufficient for me! All these versions say ordained or appointed to eternal life. I also looked at Gill and Spurgeon to see if they understood this Scripture as I do.
You don't get it, not one of these translations says God caused a person to believe. What was ordained is that the gospel would go to the Gentiles and they would believe. The last translation from Berk makes this clear.
God from eternity had ordained that the gospel would go to the Gentiles. God had ordained that Gentiles would hear the gospel and believe. But that does not mean God directly caused them to believe. God caused them to believe in the sense that he sent preachers to preach the gospel to these Gentiles, and of course the word of God is powerful and convicts a man of his sin, and enlightens him to the knowledge that he can trust in Jesus and be saved.
But all men have to believe for themselves, God does not believe for you.
I personally believe God's foreknowledge is also implied here, God knew who would believe when they heard the gospel, and so God ordained the circumstances where these persons would be given the opportunity to hear the gospel. I believe this is what Paul was saying in Acts 17;
Acts 17:26 And hath made of one blood all nations of men for to dwell on all the face of the earth,
and hath determined the times before appointed, and the bounds of their habitation;
27 That they should seek the Lord, if haply they might feel after him, and find him, though he be not far from every one of us:
This verse seems to say that God knows who will believe the gospel when they hear it, and so has determined that they be in the right place at the right time.
I believe my own experience reflects this. My family moved many times when I was young. For just a short time we lived in Panama City, Fl., My family was not religious and almost never attended church, but it just so happened we lived next to some former missionaries to Afghanistan who had a son my age. His parents asked my Dad if I could go to church with them one Sunday. I did go, and this is when I heard the gospel and accepted Jesus as my Saviour. Very soon afterwards we moved away.
Now, I think God arranged all this. He put me in just the right place at the right time so I could hear the gospel because he knew I would believe. I had always believed in God before this, but I was not saved because I had been taught a person has to be good to go to heaven. But I knew of Jesus, and I loved Jesus, and so I think God brought about the circumstances where I would hear the gospel.
And that is what Acts 13:48 is saying (IMHO), that God knew who would believe among the Gentiles, and so had ordained they would hear the gospel through Paul. So, it makes perfect sense to say "and as many as were ordained to eternal life believed".
But God does not force anyone to believe, they must believe of their own free choice.