Great question. The tension of predestination (God's sovereignty) and human free will (human choice to respond) will never be resolved in this life.

As to this verse, I'll tell you that the Greek isn't as straight forward as the English makes it sound to be. It could mean, as the English makes it sound, as if those who are saved are those whom God has chosen, a concept that is like other texts in the New Testament. It could also be, however, that Luke is merely pointing out that Gentiles were ordained for salvation just as much as Jews had been, which is very consistent with Luke's theme through the whole book. Deciding between the two choices is close to impossible, and usually people choose according to what they want to believe about it.

The point of the text, however, is not any theology of restriction on God's part, but instead the joy of people coming to the Lord, regardless of the raging debate between people about how God's works. The point is: God works, and people get saved, and let's be glad about that.

In Acts 13:48, after a rousing presentation of the Gospel message for the Gentiles, it is stated that “everyone who was appointed/ordained/chosen for eternal life believed.”

Does this text imply that God has specifically picked some people to be given eternal life and disregarded others to not be? At that rate, it would seem that not anybody can believe, but only the ones that God has picked to believe.