At the time of Acts 15 James was not in full agreement with Paul, and retained some doctrines and rituals of the Old Covenant. But - in Acts 15: 14-17 James said "Simeon hath declared how God at the first did visit the Gentiles, to take out of them a people for his name. 15. And to this agree the words of the prophets; as it is written, 16. After this I will return, and will build again the tabernacle of David, which is fallen down; and I will build again the ruins thereof, and I will set it up:
17. That the residue of men might seek after the Lord, and all the Gentiles, upon whom my name is called, saith the Lord, who doeth all these things."
James is quoting Amos 9: 11-12, "In that day will I raise up the tabernacle of David that is fallen, and close up the breaches thereof; and I will raise up his ruins, and I will build it as in the days of old:
12. That they may possess the remnant of Edom, and of all the heathen, which are called by my name, saith the LORD that doeth this."
But James says "And to this agree the words of the prophets," not just Amos. James says prophets, more than one prophet. There are several other Old Testament prophecies on the restoration of Israel, and the Christian Zionists insist these all are about the restoration of Old Covenant Israel - in a new future time when God will return to a dispensation of law and make Israel defined by the literal, physical bloodline his chosen.
Hosea 3: 5: "Afterward shall the children of Israel return, and seek the LORD their God, and David their king; and shall fear the LORD and his goodness in the latter days."
Isaiah 54: 1-3: "Sing, O barren, thou that didst not bear; break forth into singing, and cry aloud, thou that didst not travail with child: for more are the children of the desolate than the children of the married wife, saith the LORD. 2. Enlarge the place of thy tent, and let them stretch forth the curtains of thine habitations: spare not, lengthen thy cords, and strengthen thy stakes;
3. For thou shalt break forth on the right hand and on the left; and thy seed shall inherit the Gentiles, and make the desolate cities to be inhabited.”
The Old Testament prophecies on the restoration of Israel are interpreted in the New Covenant period to be about the transformation of Israel in Jesus Christ. Old Covenant Israel was remade (Jeremiah 18: 4) and translated into that spiritual house seen in I Peter 2: 5. The peculiar treasure, the kingdom of priests, and the holy nation of Old Covenant Israel of Exodus 19: 3-8 was translated into "ye" in I Peter 2: 9, "But ye are a chosen generation, a royal priesthood, an holy nation, a peculiar people; that ye should shew forth the praises of him who hath called you out of darkness into his marvellous light:"
"To translate the kingdom from the house of Saul, and to set up the throne of David over Israel and over Judah, from Dan even to Beer-sheba." II Samuel 3: 10
"Who hath delivered us from the power of darkness, and hath translated us into the kingdom of his dear Son:" Colossians 1: 13