At the Jerusalem Council, the Apostle James quoted verses 11 and 12 to support his view that the Gentiles of his day did not need to submit to circumcision and the Mosaic Law to obtain salvation or to live as Christians (Acts 15:13–21). He knew that the judgments of Israel were not yet over (cf. Matt. 24:1–22; Luke 21:5–24; Acts 1:6–7). He also knew, from this passage and others (Isa. 42:6; 60:3; Mal. 1:11), that when God restored the house of David Gentiles would have a share in that rule as Gentiles. James concluded, therefore, that Gentiles did not need to become Jews to enter into these (millennial) blessings. He did not mean that the church fulfills the promises to Israel but that since Gentiles will experience millennial blessings as Gentiles they do not need to become Jews in the church.
Tom Constable. (2003; 2003). Tom Constable’s Expository Notes on the Bible (Am 9:12). Galaxie Software.
Who?
The actual words of Jame's speech in Acts 15
James answered, saying, Brethren, hearken unto me: 14Symeon hath rehearsed how first God visited the Gentiles, to take out of them a people for his name. 15And to this agree the words of the prophets; as it is written, 16After these things I will return, And I will build again the tabernacle of David, which is fallen; And I will build again the ruins thereof, And I will set it up: 17That the residue of men may seek after the Lord, And all the Gentiles,
American Standard Version. 1995 (Ac 15:13). Oak Harbor, WA: Logos Research Systems, Inc.
A bit more explanation...
15:15 James reminded his hearers that the Old Testament prophets supported the salvation of Gentiles apart from Judaism. Note that James did not say the salvation of Gentiles then was the fulfillment of these prophecies. He said the prophets’ predictions of future Gentile salvation harmonized with the present salvation of Gentiles apart from Judaism (cf. 2:16).614 James then quoted Amos 9:11–12 as a representative prophecy.615 Neither Amos nor any other prophet said Gentiles had to become Jews to enjoy the blessings of salvation (cf. Rom. 11:12).
Tom Constable. (2003; 2003). Tom Constable's Expository Notes on the Bible (Ac 15:15). Galaxie Software.
Please note that James does not say that the prophecy was fulfilled by the Gentile conversion. Just that the Gentile conversion was in harmony with what the prophet wrote in Amos.
best,
ampu