Short answer
No, not on the Gospels' own account. In the Bible, God (﷾) promises that David's throne will stand forever through the line of his son Solomon. That throne ended in the sixth century BC with King Zedekiah, and no king of David's line has ruled since. Christians say Jesus fulfils the promise, but every route fails. Luke traces Jesus through Nathan, not Solomon. Matthew goes through Solomon, but also through King Jechonias, and Jeremiah records God (﷾) saying that no man of his seed would prosper on David's throne. Both Gospels also say Joseph was not Jesus' father by birth, so neither line is his by blood. Islam honours Jesus as al-Masih, the Messiah, and a messenger to Israel. It gives him no claim to David's throne.
Full explanation
The promise Christians see fulfilled
The Bible says God (﷾) promised David that his throne would be established for ever. In Luke, one of the New Testament Gospels, the angel tells Mary that God (﷾) will give her son "the throne of his father David".
“And thine house and thy kingdom shall be established for ever before thee: thy throne shall be established for ever.”
“But I will settle him in mine house and in my kingdom for ever: and his throne shall be established for evermore.”
“He shall be great, and shall be called the Son of the Highest: and the Lord God shall give unto him the throne of his father David: And he shall reign over the house of Jacob for ever; and of his kingdom there shall be no end.”
The Bible ties the promise to Solomon
Chronicles names the son to whom the everlasting throne was promised: Solomon. David says God (﷾) chose Solomon, out of all his sons, to sit on the throne.
“Behold, a son shall be born to thee, who shall be a man of rest; and I will give him rest from all his enemies round about: for his name shall be Solomon, and I will give peace and quietness unto Israel in his days. He shall build an house for my name; and he shall be my son, and I will be his father; and I will establish the throne of his kingdom over Israel for ever.”
“And of all my sons, (for the Lord hath given me many sons,) he hath chosen Solomon my son to sit upon the throne of the kingdom of the Lord over Israel.”
The throne ended with Zedekiah
Zedekiah was the last king of Judah. The Babylonians killed his sons in front of him, blinded him and took him to Babylon. After Jerusalem fell in 586 BC, no king of David's line ever sat on the throne again.
“And they slew the sons of Zedekiah before his eyes, and put out the eyes of Zedekiah, and bound him with fetters of brass, and carried him to Babylon.”
Route one: Luke's family tree
Luke traces Jesus back to David through Nathan, another son of David, not through Solomon. The promise of the throne was never made to Nathan's line.
“Which was the son of Melea, which was the son of Menan, which was the son of Mattatha, which was the son of Nathan, which was the son of David,”
“Behold, a son shall be born to thee, who shall be a man of rest; and I will give him rest from all his enemies round about: for his name shall be Solomon, and I will give peace and quietness unto Israel in his days. He shall build an house for my name; and he shall be my son, and I will be his father; and I will establish the throne of his kingdom over Israel for ever.”
Route two: Matthew's family tree
Matthew traces Jesus through Solomon, but his line passes through King Jechonias, also called Coniah. In Jeremiah, God (﷾) is recorded saying of this king that no man of his seed would prosper sitting on David's throne.
“And Josias begat Jechonias and his brethren, about the time they were carried away to Babylon: And after they were brought to Babylon, Jechonias begat Salathiel; and Salathiel begat Zorobabel;”
“As I live, saith the Lord, though Coniah the son of Jehoiakim king of Judah were the signet upon my right hand, yet would I pluck thee thence;”
“Thus saith the Lord, Write ye this man childless, a man that shall not prosper in his days: for no man of his seed shall prosper, sitting upon the throne of David, and ruling any more in Judah.”
Route three: Joseph was not his father
Both family trees end with Joseph. Yet Matthew says Mary was with child by the Holy Spirit before she and Joseph came together. Luke says Jesus was only "supposed" to be Joseph's son. On the Gospels' own account, neither line reaches Jesus by birth.
“And Jacob begat Joseph the husband of Mary, of whom was born Jesus, who is called Christ.”
“Now the birth of Jesus Christ was on this wise: When as his mother Mary was espoused to Joseph, before they came together, she was found with child of the Holy Ghost.”
“And Jesus himself began to be about thirty years of age, being (as was supposed) the son of Joseph, which was the son of Heli,”
In the Bible, God (﷾) promises that David's throne will stand for ever, through Solomon's line.
What happened to the throne?
It ended in the sixth century BC with King Zedekiah. No king of David's line has ruled since.
The Christian answer
Jesus fulfils the promise. The angel told Mary that God (﷾) would give him “the throne of his father David”.
Route one: Luke's family tree
Luke traces Jesus through Nathan, another son of David, not through Solomon. The promise was never made to Nathan's line.
“It is Mary's line” Luke names Joseph, not Mary. And any line through Nathan still misses Solomon's throne.
Route two: Matthew's family tree
Matthew goes through Solomon, but also through King Jechonias. Jeremiah records God (﷾) saying “no man of his seed shall prosper, sitting upon the throne of David”.
Route three: through Joseph
Both Gospels say Joseph was not Jesus' father by birth, so neither family tree reaches Jesus by blood.
“Joseph adopted him” Joseph was of Jechonias' seed, which Jeremiah says God (﷾) barred from the throne. Joseph had no right to pass on, and adoption cannot give Jesus a right Joseph did not have.
Every route fails. The Qur'an honours Jesus as al-Masih, the Messiah, and a messenger to Israel. It gives him no claim to David's throne.
What the Qur'an says instead
The Qur'an calls Jesus al-Masih, the Messiah, honoured in this world and the next, and a messenger to the Children of Israel. The title is God's (﷾) gift to him (3:45), and it does not rest on David's throne. The Qur'an agrees that Mary bore Jesus although no man had touched her (19:20). Islam teaches that Jesus will return as a just ruler (Bukhari 3448) and lead the Muslims as one of them (Muslim 155). That is God's (﷾) appointment, not a claim through David's line.
“Behold! the angels said: "O Mary! Allah giveth thee glad tidings of a Word from Him: his name will be Christ Jesus, the son of Mary, held in honour in this world and the Hereafter and of (the company of) those nearest to Allah;”
“"And (appoint him) a messenger to the Children of Israel, (with this message): "'I have come to you, with a Sign from your Lord, in that I make for you out of clay, as it were, the figure of a bird, and breathe into it, and it becomes a bird by Allah's leave: And I heal those born blind, and the lepers, and I quicken the dead, by Allah's leave; and I declare to you what ye eat, and what ye store in your houses. Surely therein is a Sign for you if ye did believe;”
“She said: "How shall I have a son, seeing that no man has touched me, and I am not unchaste?"”
Abu Huraira reports that the Prophet (ﷺ) swore that the son of Mary will soon descend among the Muslims as a just ruler. He will break the cross, kill the swine and end the jizya, and wealth will be so plentiful that no one will accept it. Abu Huraira then recited Qur'an 4:159: none of the People of the Book but will believe in him before his death.
In 155a Abu Huraira reports that the Prophet (ﷺ) swore the son of Mary will soon descend as a just judge, break the cross, kill the swine and end the jizya. In 155f the Prophet asks how it will be when the son of Mary descends and leads the Muslims as one of them; the narrator Ibn Abi Dhi'b explains that he will lead them by the Book of their Lord and the Sunnah of their Prophet.
Verse by verse
Luke 3:23 — Gospel of Luke, chapter 3, verse 23
“And Jesus himself began to be about thirty years of age, being (as was supposed) the son of Joseph, which was the son of Heli,”
Cited for
Luke's family tree is really Mary's. Heli was Mary's father, so Jesus descends from David by blood.
The response
Luke names Joseph, not Mary, and calls Heli Joseph's father. Mary appears nowhere in the list. Even if it were her line, it runs through Nathan, and the throne was promised to Solomon's line.
Where it lands — Luke's line cannot carry Solomon's throne, whoever it belongs to.
1 Chronicles 28:7 — Ketuvim, Divrei HaYamim Aleph, chapter 28, verse 7
“Moreover I will establish his kingdom for ever, if he be constant to do my commandments and my judgments, as at this day.”
Cited for
The promise to Solomon's line was conditional. Solomon's line lost it by disobedience, so the throne can pass through Nathan, who is also David's seed.
The response
In God's (﷾) promise about the son who would build His house, He says: "If he commit iniquity, I will chasten him". He adds: "But my mercy shall not depart away from him, as I took it from Saul." Sin brings chastening, not the loss of the throne. God (﷾) also said of Solomon, "I will establish the throne of his kingdom over Israel for ever." If his line could lose it after all, that word, as the Bible gives it, failed. And no verse ever passes the promise to Nathan.
Where it lands — Either the promise stayed with Solomon's line, which rules out Luke, or the promise as written failed; neither gives the throne to Jesus.
Matthew 1:16 — Gospel of Matthew, chapter 1, verse 16
“And Jacob begat Joseph the husband of Mary, of whom was born Jesus, who is called Christ.”
Cited for
Joseph adopted Jesus, so Jesus holds David's throne by legal right through Joseph's royal line.
The response
On Matthew's tree, Joseph is of Jechonias' seed, and Jeremiah records that God (﷾) barred that seed from David's throne. So Joseph had no right to the throne to pass on. Adoption cannot give Jesus a right that Joseph himself did not have.
Where it lands — The legal line through Joseph is the one line God (﷾) closed to the throne.
Haggai 2:23 — Nevi'im, Haggai, chapter 2, verse 23
“In that day, saith the Lord of hosts, will I take thee, O Zerubbabel, my servant, the son of Shealtiel, saith the Lord, and will make thee as a signet: for I have chosen thee, saith the Lord of hosts.”
Cited for
God lifted the curse. Jechonias' grandson Zerubbabel is made God's signet, reversing the words spoken over Coniah.
The response
Haggai calls Zerubbabel a signet, but it says nothing of a throne. Zerubbabel was governor of Judah under Persia, not king. No descendant of Jechonias ever ruled on David's throne, just as Jeremiah said.
Where it lands — Jeremiah's words about the throne were never reversed; history kept them.
Luke 1:32-33 — Gospel of Luke, chapter 1, verses 32-33
“He shall be great, and shall be called the Son of the Highest: and the Lord God shall give unto him the throne of his father David: And he shall reign over the house of Jacob for ever; and of his kingdom there shall be no end.”
Cited for
Jesus took David's throne when he rose to heaven. He reigns there now, and his kingdom has no end.
The response
The angel promised "the throne of his father David" and a reign "over the house of Jacob". David's throne was an earthly throne in Jerusalem over Israel. Jesus has not sat on it, and Israel did not take him as king. Moving the throne to heaven changes the promise. It does not fulfil it. And the question of the line stays the same. Whoever sits on David's throne must be David's heir through Solomon. Neither Gospel line gives Jesus that.
Where it lands — A throne in heaven is not the throne of David over the house of Jacob.
Perspectives
Mainstream Christian
Jesus is the promised Son of David. He has the legal right to the throne through Joseph, his father by law. He reigns now and will reign for ever, as the angel told Mary.
The angel in Luke promises that God will give Jesus the throne of his father David. His kingdom, the angel says, will have no end.
Rebuttal
Luke does record the promise. But the angel spoke of "the throne of his father David" and a reign "over the house of Jacob". That was an earthly throne in Jerusalem. Jesus has not sat on it. On the Bible's own terms, its heir must come through Solomon, and neither Gospel line gives Jesus that.
Luke's family tree is Mary's line, so Jesus descends from David by blood through Nathan.
Rebuttal
Luke names Joseph, not Mary, and calls Heli Joseph's father. Mary appears nowhere in the list. Even if it were her line, it runs through Nathan, and the throne was promised to Solomon's line.
The Islamic perspective
The throne the Bible promised to David's line ended with Zedekiah, and the Gospels cannot give it to Jesus by any route. Jesus did not need it. The Qur'an honours him as al-Masih, the Messiah, and a messenger of God (﷾), not as the heir to David's throne.
The Bible ties the promise of an everlasting throne to Solomon's line.
The throne of David ended with Zedekiah, and no king of David's line reigned after the fall of Jerusalem.
Luke's line runs through Nathan, not Solomon.
Matthew's line runs through Jechonias, whose seed Jeremiah says God (﷾) barred from prospering on David's throne.
Matthew and Luke both say Joseph was not Jesus' father by birth.
The Qur'an honours Jesus as al-Masih, the Messiah, and a messenger to Israel, not as the heir to David's throne.