Short answer
No. A birth without a father is a miracle, and a miracle does not make a man God. Adam had no father and no mother. Luke's Gospel, a New Testament book about Jesus, calls Adam "the son of God". Eve had no mother. If a fatherless birth made Jesus God, Adam would have the better claim. Yet no Christian says Adam is God. Christian teachers themselves do not rest Jesus' divinity on the birth alone. The Bible also calls Israel God's son, so the title does not mean "God". The Qur'an gives the answer: Jesus is like Adam. God (﷾) made him by saying "Be!", and he was. Muslims believe in the virgin birth with all their hearts. For Islam, it shows the power of God (﷾), not that Jesus is God.
Full explanation
Muslims believe in the virgin birth
The Qur'an tells the story of the virgin birth in chapters 3 and 19. Mary asks how she can have a son when no man has touched her. She is told that God (﷾) creates what He wills by saying "Be". The Qur'an honours Mary as chosen above the women of all nations. It calls her son a sign and a mercy.
“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;”
“She said: "O my Lord! How shall I have a son when no man hath touched me?" He said: "Even so: Allah createth what He willeth: When He hath decreed a plan, He but saith to it, 'Be,' and it is!”
“She said: "How shall I have a son, seeing that no man has touched me, and I am not unchaste?"”
“He said: "So (it will be): Thy Lord saith, 'that is easy for Me: and (We wish) to appoint him as a Sign unto men and a Mercy from Us': It is a matter (so) decreed."”
“Behold! the angels said: "O Mary! Allah hath chosen thee and purified thee- chosen thee above the women of all nations.”
What Christians say
In Luke 1, an angel tells Mary that the Holy Ghost (the Holy Spirit) will come upon her. Because of this, her child will be called the Son of God. Christians teach that the eternal Son took flesh in Mary's womb. Many Christians read the fatherless birth as a sign that Jesus is God's Son in a way no one else is.
“Then said Mary unto the angel, How shall this be, seeing I know not a man? And the angel answered and said unto her, The Holy Ghost shall come upon thee, and the power of the Highest shall overshadow thee: therefore also that holy thing which shall be born of thee shall be called the Son of God.”
Adam had neither father nor mother
Genesis, the first book of the Bible, says God (﷾) formed Adam from the dust and breathed life into him. So Adam had no father and no mother. Eve had no mother, because God (﷾) made her from a rib taken from Adam. Jesus at least had a mother. Two Gospels give him a family tree, and Luke's tree ends by calling Adam "the son of God". Yet no Christian says Adam or Eve was God.
“And the Lord God formed man of the dust of the ground, and breathed into his nostrils the breath of life; and man became a living soul.”
“And the Lord God caused a deep sleep to fall upon Adam, and he slept: and he took one of his ribs, and closed up the flesh instead thereof; And the rib, which the Lord God had taken from man, made he a woman, and brought her unto the man.”
“Which was the son of Enos, which was the son of Seth, which was the son of Adam, which was the son of God.”
"Son of God" in the Bible's own language
The Bible calls others God's sons without making them divine. God (﷾) calls the nation of Israel "my son, even my firstborn". In Luke, Jesus says that those raised from the dead are "the children of God". Luke calls Adam the son of God because God (﷾) made him directly. So the title marks nearness to God (﷾), not a share in His nature.
“And thou shalt say unto Pharaoh, Thus saith the Lord, Israel is my son, even my firstborn:”
“Neither can they die any more: for they are equal unto the angels; and are the children of God, being the children of the resurrection.”
“Which was the son of Enos, which was the son of Seth, which was the son of Adam, which was the son of God.”
The Qur'an's answer in one sentence
The Qur'an answers the fatherless-birth argument by pointing to Adam. The likeness of Jesus before God (﷾) is the likeness of Adam, whom God (﷾) created from dust by the word "Be". The Qur'an adds that it is not fitting for God (﷾) to take a son. When He decides a matter, He only says "Be".
“The similitude of Jesus before Allah is as that of Adam; He created him from dust, then said to him: "Be". And he was.”
“It is not befitting to (the majesty of) Allah that He should beget a son. Glory be to Him! when He determines a matter, He only says to it, "Be", and it is.”
"His Word" means His command
The Qur'an calls Jesus a Word from God (﷾). Muslim scholars explain, from verses 3:47 and 3:59, that this word is God's (﷾) command "Be". That command brought Jesus into being without a father. The same verse that calls him God's (﷾) Word also calls him no more than a messenger of Allah (﷾). It forbids saying "Trinity", the Christian teaching of one God in three persons.
“O People of the Book! Commit no excesses in your religion: Nor say of Allah aught but the truth. Christ Jesus the son of Mary was (no more than) a messenger of Allah, and His Word, which He bestowed on Mary, and a spirit proceeding from Him: so believe in Allah and His messengers. Say not "Trinity": desist: it will be better for you: for Allah is one Allah: Glory be to Him: (far exalted is He) above having a son. To Him belong all things in the heavens and on earth. And enough is Allah as a Disposer of affairs.”
“She said: "O my Lord! How shall I have a son when no man hath touched me?" He said: "Even so: Allah createth what He willeth: When He hath decreed a plan, He but saith to it, 'Be,' and it is!”
Verse by verse
Luke 1:34-35 — Gospel of Luke, chapter 1, verses 34-35
“Then said Mary unto the angel, How shall this be, seeing I know not a man? And the angel answered and said unto her, The Holy Ghost shall come upon thee, and the power of the Highest shall overshadow thee: therefore also that holy thing which shall be born of thee shall be called the Son of God.”
Cited for
Christians read the angel's words as linking the virgin conception to Jesus being the divine Son of God.
The response
Luke himself calls Adam "the son of God" two chapters later, because God (﷾) made him directly. By Luke's own usage, a child brought into being by God's (﷾) power is called God's son. That is a title of honour, not a claim to be God.
Where it lands — The same Gospel uses the same title for Adam, so the birth does not show that Jesus is God.
Qur'an 4:171 — Surat al-Nisa (The Women), verse 171
“O People of the Book! Commit no excesses in your religion: Nor say of Allah aught but the truth. Christ Jesus the son of Mary was (no more than) a messenger of Allah, and His Word, which He bestowed on Mary, and a spirit proceeding from Him: so believe in Allah and His messengers. Say not "Trinity": desist: it will be better for you: for Allah is one Allah: Glory be to Him: (far exalted is He) above having a son. To Him belong all things in the heavens and on earth. And enough is Allah as a Disposer of affairs.”
Cited for
Some Christians argue that the Qur'an itself calls Jesus God's "Word" and "a spirit from Him", and so points to his divinity.
The response
The same verse calls Jesus no more than a messenger of Allah (﷾). It forbids saying "Trinity", and it says God (﷾) is far above having a son. Two other verses show how Jesus came to be: by God's (﷾) command "Be". That is what Muslim scholars take the Word to mean. And "a spirit from Him" makes no one divine: the Qur'an says God (﷾) breathed of His spirit into Adam too.
Where it lands — Read whole, the verse denies what the argument tries to draw from it.
Perspectives
Mainstream Christian
Christians believe the virgin birth is how the eternal Son of God became a man. They do not rest his divinity on the birth alone. But they see it as the sign that he is God's Son in a unique sense.
Luke's angel speaks of the Holy Ghost coming upon Mary. Because of this, he says, the child will be called the Son of God.
Rebuttal
Luke does link the Holy Ghost to the title. But two chapters later Luke calls Adam "the son of God", and Exodus calls Israel God's (﷾) "son". In the Bible the title shows closeness to God (﷾), not that someone is God.
The fatherless conception marks Jesus as God's Son in a way that no other person is.
Rebuttal
The birth is a real miracle, and Muslims believe in it. But Adam had no father and no mother, and Luke still calls him God's son. A miracle shows God's (﷾) power, not that the one born is God.
The Islamic perspective
Jesus was born of a virgin by God's (﷾) command, as a sign and a mercy. That makes him a miracle and a great messenger, not God and not God's son. Adam, made with no father and no mother, is the proof.
The Qur'an says that Mary conceived Jesus though no man had touched her.
Adam had no father and no mother, and Luke still calls him "the son of God".
If a fatherless birth made someone God, Adam would have a stronger claim than Jesus. So the argument proves too much.
The Bible calls Israel God's son, and calls those raised from the dead God's (﷾) children. It does not make them divine.
The Qur'an teaches that Jesus is like Adam, created by God's (﷾) word "Be". It teaches that God (﷾) does not take a son.