Yes, it is prohibited and all the Muslim scholars are unanimously agreed that it is not permissible for a Muslim woman to marry a non-Muslim, whether he is Jewish, Christian or any other, because Allah said in the Quran what might be translated as:
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, evidence, verses, lessons, signs, revelations, etc.) clear to mankind that they may remember)
(…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)
The above verses indicate that the main rule in Islam on interfaith marriage is impermissibility for both genders.
Then, Allah gave an exception to the main rule and allowed Muslim men to marry certain women of ahlu alkitab ( Christian and Jewish women ).
Allah may He be exalted, says (interpretation of the meaning):
(… And [lawful in marriage are] chaste women from among the believers and chaste women from among those who were given the Scripture before you, when you have given them their due compensation, desiring chastity, not unlawful sexual intercourse or taking [secret] lovers. (Al-Ma’idah, 5).
From the above verse, we learn that even with the exception of the rule, Allah made it very clear that this exception only applies to certain criteria of women who are chaste.
When Muslim women ask why so and how come the same ruling does not apply to Muslim men, we usually share the following fiqh foundational principles of jurisprudence that could be applied to many other matters in our daily Islamic practices:
On divisive matters of fiqh (Islamic rulings), it is very important to come back to the usul (foundational principles of jurisprudence – i.e. how one arrives at rulings), which set out the rules of engagement of the text. Otherwise, anyone will deem themselves qualified to engage the primary sources and they will lose all meaning because each will see what they want to see.
– Understanding the hikmah (Divine Wisdom) of a ruling is different from a text that establishes the ruling itself (hukm). E.g., knowing the wisdom behind 5 prayers instead of 6 or 4 is different from an authoritative text that establishes they are 5.
– We firmly believe there is always wisdom behind every divine legislation Allah mandated but not necessarily shared with His servants. In this case, the wisdom that has been outlined by scholars as to why not a Muslim woman can marry a non-Muslim man is all indirectly inferred and not explicitly stated in the primary text of the Quran or Sunnah. Therefore while they may be helpful in developing a greater appreciation, they are not helpful in establishing halal or haram in this case.
– That is: any explanation of “why” where it is not expressly mentioned in the text, is a person’s honest reasoning. This should be differentiated from when Allah tells us part of the express wisdom in a ruling, e.g. the proper fast of Ramadan connected to the achievement of taqwa.
– That being said: the hukm among the consensus of Muslim jurists based on the texts of the Quran and Sunnah has been the prohibition of such marriages.