James (the speaker) refers to Amos 9, one of many OT passages that express the divine promise that God will restore the fortunes of Israel after their punishment, a result of their rejection of the God of Abraham in deference to idols. In particular, Amos is a prophet to the northern 10 tribes, whose recovery from exile was never fulfilled in any "literal" manner.
And yet, the God who never lies promises that he will "raise up the tabernacle of David which has fallen down... that they may possess the remnant of Edom and all the Gentiles who are called by my name." The interpretation James offers is along this line: the throne of David IS NOW restored. The king came; he won; he sat down at the right hand of Power. The kingdom has been reestablished in righteousness. And so, as corollary to this outstanding fact, we are now also seeing the promised evidence of the king's success as the king begins to "possess" his claims among the Gentiles. Compare to the same in Is.49:6.
According to Amos, there are Gentiles who are elect, "called by my name." So, it is the special interest of the conquering King to lay claim to them also, and to incorporate them (properly so) into his kingdom. This, James recognizes as the reality. So then the church, as the embassy of Christ in the world, shall recognize as Christ's those whom Christ recognizes by his pouring out on them (no less than on Jews/Israelites) the inheritance his Holy Spirit. They possess his mark of citizenship--who are men to demand more?
