There are a number of conditions of the promised Davidic kingdom that began to be fulfilled even as Christ was born. First, it is evident that in the birth of Christ, Israel’s Davidic Savior had arrived. When Simeon saw the Christ child in the temple he said,
Blessed be the Lord, the God of Israel; for he hath visited and wrought redemption for his people, And hath raised up a horn of salvation for us in the house of his servant David (Luke 1:69). Simeon realized that he could finally die in peace, for he had seen the day long promised.
Furthermore, after the ascension of Christ, Paul recounted in the synagogue in Antioch how God had raised up David to be Israel’s king;
to whom he also bare witness and said, I have found David the son of Jesse a man after my own heart, who shall do all my will. Paul then added:
of this man’s seed hath God according to promise brought unto Israel a savior, Jesus (Acts 13:22-23). God’s promise to Israel had been fulfilled. The long awaited Davidic Savior had arrived.
Not only had Israel’s Savior arisen out of the house of David, but in that Savior that house was also itself being rebuilt. At the Jerusalem Council, James explained in this way God’s obvious decision to begin calling Gentiles into the faith:
After 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: That the residue of men may seek after the Lord, And all the Gentiles, upon whom my name is called (Acts 15:16-17). The fact that the Gentiles were putting their trust in Christ was proof, to James at least, that David’s house was being restored.
So it appears that, as the New Testament was still being written, the promises of a Davidic Savior and the restoration of the David's house were already being fulfilled. Do these passages prove beyond doubt that Christ is now reigning on the throne of David? Not by themselves, no. They are, however, important introductory indications that whatever God had promised concerning David and the kingdom of his Descendant, that program had clearly begun to unfold.