Objection 1. It would seem that Christ did not take flesh of the seed of David. For Matthew, in tracing the genealogy of Christ, brings it down to Joseph. But Joseph was not Christ's father, as shown above (28, 1, ad 1,2). Therefore it seems that Christ was not descended from David.
Objection 2. Further, Aaron was of the tribe of Levi, as related Exodus 6. Now Mary the Mother of Christ is called the cousin of Elizabeth, who was a daughter of Aaron, as is clear from Luke 1:5-36. Therefore, since David was of the tribe of Juda, as is shown Matthew 1, it seems that Christ was not descended from David.
Objection 3. Further, it is written of Jechonias (Jeremiah 22:30): "Write this man barren... for there shall not be a man of his seed that shall sit upon the throne of David." Whereas of Christ it is written (Isaiah 9:7): "He shall sit upon the throne of David." Therefore Christ was not of the seed of Jechonias: nor, consequently, of the family of David, since Matthew traces the genealogy from David through Jechonias.
On the contrary, It is written (Romans 1:3): "Who was made to him of the seed of David according to the flesh."
I answer that, Christ is said to have been the son especially of two of the patriarchs, Abraham and David, as is clear from Matthew 1:1. There are many reasons for this. First to these especially was the promise made concerning Christ. For it was said to Abraham (Genesis 22:18): "In thy seed shall all the nations of the earth be blessed": which words the Apostle expounds of Christ (Galatians 3:16): "To Abraham were the promises made and to his seed. He saith not, 'And to his seeds' as of many; but as of one, 'And to thy seed,' which is Christ." And to David it was said (Psalm 131:11): "Of the fruit of thy womb I will set upon thy throne." Wherefore the Jewish people, receiving Him with kingly honor, said (Matthew 21:9): "Hosanna to the Son of David."
A second reason is because Christ was to be king, prophet, and priest. Now Abraham was a priest; which is clear from the Lord saying unto him (Genesis 15:9): "Take thee [Vulgate: 'Me'] a cow of three years old," etc. He was also a prophet, according to Genesis 20:7: "He is a prophet; and he shall pray for thee." Lastly David was both king and prophet.
A third reason is because circumcision had its beginning in Abraham: while in David God's election was most clearly made manifest, according to 1 Samuel 13:14: "The Lord hath sought Him a man according to His own heart." And consequently Christ is called in a most special way the Son of both, in order to show that He came for the salvation both of the circumcised and of the elect among the Gentiles.
Reply to Objection 1. Faustus the Manichean argued thus, in the desire to prove that Christ is not the Son of David, because He was not conceived of Joseph, in whom Matthew's genealogy terminates. Augustine answered this argument thus (Contra Faust. xxii): "Since the same evangelist affirms that Joseph was Mary's husband and that Christ's mother was a virgin, and that Christ was of the seed of Abraham, what must we believe, but that Mary was not a stranger to the family of David: and that it is not without reason that she was called the wife of Joseph, by reason of the close alliance of their hearts, although not mingled in the flesh; and that the genealogy is traced down to Joseph rather than to her by reason of the dignity of the husband? So therefore we believe that Mary was also of the family of David: because we believe the Scriptures, which assert both that Christ was of the seed of David according to the flesh, and that Mary was His Mother, not by sexual intercourse but retaining her virginity." For as Jerome says on Matthew 1:18: "Joseph and Mary were of the same tribe: wherefore he was bound by law to marry her as she was his kinswoman. Hence it was that they were enrolled together at Bethlehem, as being descended from the same stock."
Reply to Objection 2. Gregory of Nazianzum answers this objection by saying that it happened by God's will, that the royal family was united to the priestly race, so that Christ, who is both king and priest, should be born of both according to the flesh. Wherefore Aaron, who was the first priest according to the Law, married a wife of the tribe of Juda, Elizabeth, daughter of Aminadab. It is therefore possible that Elizabeth's father married a wife of the family of David, through whom the Blessed Virgin Mary, who was of the family of David, would be a cousin of Elizabeth. or conversely, and with greater likelihood, that the Blessed Mary's father, who was of the family of David, married a wife of the family of Aaron.
Again, it may be said with Augustine (Contra Faust. xxii) that if Joachim, Mary's father, was of the family of Aaron (as the heretic Faustus pretended to prove from certain apocryphal writings), then we must believe that Joachim's mother, or else his wife, was of the family of David, so long as we say that Mary was in some way descended from David.
Reply to Objection 3. As Ambrose says on Luke 3:25, this prophetical passage does not deny that a posterity will be born of the seed of Jechonias. And so Christ is of his seed. Neither is the fact that Christ reigned contrary to prophecy, for He did not reign with worldly honor; since He declared: "My kingdom is not of this world."
Source: Summa Theologica (New Advent)