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You did not explain what this man has done such that he deserves to be regarded by you as a mushrik. But whatever the case, if it is in fact proven that he is a mushrik who ascribes partners to Allah, then it is not permissible for a Muslim woman to marry him, because Allah, may He be glorified, has commanded the Muslims not to give their daughters in marriage to people of shirk, and He has explained the reason for this prohibition, which is that marriage of a Muslim woman to a mushrik opens the door to her of confusion about religion and apostasy from Islam. Allah, may He be glorified, says (interpretation of the meaning):
“And give not (your daughters) in marriage to Al-Mushrikoon till they believe (in Allah Alone) and verily, a believing slave is better than a (free) Mushrik (idolater, etc.), even though he pleases you. Those (Al-Mushrikoon) invite you to the Fire, but Allah invites (you) to Paradise and Forgiveness by His Leave, and makes His Ayat (proofs, evidences, verses, lessons, signs, revelations, etc.) clear to mankind that they may remember”
[al-Baqarah 2:221].
Al-Baghawi (may Allah have mercy on him) said:
“And give not (your daughters) in marriage to Al-Mushrikoon till they believe (in Allah Alone)” – there is consensus on this point: it is not permissible for a Muslim woman to marry a mushrik man.
“and verily, a believing slave is better than a (free) Mushrik (idolater, etc.), even though he pleases you.”
“Those” i.e., the mushrikeen or polytheists
“invite you to the Fire” i.e., to deeds that lead one to Hell.
Al-Qurtubi (may Allah have mercy on him) said in his Tafseer (3/72):
“And give not (your daughters) in marriage to Al-Mushrikoon” means do not give a Muslim woman in marriage to a mushrik. The ummah is unanimously agreed that a mushrik should not marry a believing woman under any circumstances, because that undermines Islam. End quote.
Allah, may He be glorified and exalted, says (interpretation of the meaning):
“O you who believe! When believing women come to you as emigrants, examine them, Allah knows best as to their Faith, then if you ascertain that they are true believers, send them not back to the disbelievers, they are not lawful (wives) for the disbelievers nor are the disbelievers lawful (husbands) for them”
[al-Mumtahanah 60:10].
Ibn Katheer (may Allah have mercy on him) said in his Tafseer (8/93): “they are not lawful (wives) for the disbelievers nor are the disbelievers lawful (husbands) for them”. This verse is the one which forbade Muslim women as wives for mushrik men. At the beginning of Islam it had been permissible for a mushrik man to marry a believing woman. End quote.
Ash-Shawkaani said in Fath al-Qadeer (5/256): The sentence “they are not lawful (wives) for the disbelievers nor are the disbelievers lawful (husbands) for them” explained the reason why they should not be turned back. This indicates that a believing woman is not permissible for a disbelieving man, and that if a woman becomes Muslim, she must be separated from her (mushrik) husband. End quote.
You should have been careful with your daughter and chosen from her potential suitors one who was religiously-committed and of good character, and you should have enquired about him and found out more about him. But you were careless and heedless about that, to the extent that this happened and she married that man who is a mushrik, according to what you say. This marriage was undoubtedly invalid from the moment it occurred, according to Muslim consensus, so what your daughter must do is leave this man immediately, without talaaq (divorce), because talaaq can only result from a valid marriage, but this marriage was invalid from the outset.
Shaykh ‘Abd al-‘Azeez ibn Baaz (may Allah have mercy on him) was asked: A man married a Muslim woman, then it became clear that the man was a kaafir. What is the ruling?
He replied:
if it is proven that the man mentioned was a kaafir at the time of the marriage contract, and the woman was a Muslim, then the marriage is invalid, because it is not permissible, according to Muslim consensus, for a kaafir to marry a Muslim woman, because Allah, may He be glorified, says (interpretation of the meaning):
“And give not (your daughters) in marriage to Al-Mushrikoon till they believe (in Allah Alone)”
[al-Baqarah 2:221].
“… then if you ascertain that they are true believers, send them not back to the disbelievers, they are not lawful (wives) for the disbelievers nor are the disbelievers lawful (husbands) for them”
[al-Mumtahanah 60:10].
End quote from Fataawa Islamiyyah (3/230)
In a statement of the Islamic Fiqh Council it says:
Marriage of a kaafir to a Muslim woman is haraam and is not permissible, according to scholarly consensus, and there is no doubt about that because of what is stated in the shar‘i texts.
End quote from Fataawa Islamiyyah (3/231)
What you must do is repent to Allah, may He be glorified, from this heedlessness, and you should hasten to rectify the sitution with regard to your daughter.
All of this applies if what you meant by saying that this man is a mushrik is that it became clear to you that he follows a religion other than Islam, such as the Babi, Baha’i, Qadiani, Bohra and other sects that are outside the religion of Islam.
But if what you meant is that he is a Muslim, in basic terms, but he does some of the acts of the mushrikeen that are not obvious to Muslims like him, especially in countries where knowledge of Islam and the Sunnah is weak, then in this case he should be told about what the Messenger taught about such matters, and the religion of Allah should be explained to him, and proof from the Sunnah should be established for him. Then if he repents from his ways and gives up, there is nothing wrong with your daughter remaining married to him.
But if he insists on following that way, then the ruling is as described above, that she must separate from him.
You should understand that according to Islam, it is essential to have a wali (guardian) for marriage, and it is not permissible for a woman to give herself or another woman in marriage. This has been explained in detail in fatwa no. 104852
And Allah knows best.