If a non-Muslim woman, with a non-religious but spiritual upbringing, is a monotheist and is not an idol-worshipper, nor harbors faith in multiple gods, is it wrong for a Muslim man to marry her? 

Please advise.

In the name of Allah, the Most Merciful, the Grantor of Mercy

All praise belongs to Allah, and may prayers and peace be upon Allah’s final Messenger.

The advice of the Prophet (Allah's blessings and peace be upon him) concerning marriage is:

تنكح المرأة لأربع لمالها ولحسبها وجمالها ولدينها فاظفر بذات الدين تربت يداك"

"A woman is sought after for marriage for [any of] four reasons: her wealth, her lineage (status), her beauty, and her deen; so win the religious one [and] may you prosper." [Agreed upon, from Abi Hurairah]

Marrying a righteous Muslim woman has great benefits for one’s own religious commitment and the righteous upbringing of their children.

It is allowed for a Muslim to marry a woman who is Muslim, Christian, or Jewish. As for marrying the women of the People of the Book (i.e., Jewish or Christian), some of the Companions, including ’Umar ibn al-Khattab, prevented – without calling it haram (except for ibn ’Umar who considered it haram) - Muslim men from marrying non-Muslim women, because this may put the righteous Muslim women at a disadvantage. (They can’t marry men who aren’t Muslim.) He also feared that the Muslims may marry non-Muslim women who are not chaste. 

It is also a bigger concern for Muslims to marry non-Muslim women of the People of the Book in a land where the prevailing laws are un-Islamic, since they will be subject to these laws, and their children may be taken from them in case of conflict. 

Many scholars limited the meaning of the verse in which Allah allowed the marriage of Muslim men to non-Muslim women of the People of the Book to situations where Islam is the predominant religion, and stated that it is prohibited to marry a Christian or Jewish woman whose people are at war with the Muslims. 

Based on all the above, though we can't say it is haram, it is strongly discouraged for Muslim men to marry non-Muslim women in lands where Muslims are the minority.

If the woman you are talking about is not Muslim, Christian, or Jewish, then it is not permissible to marry her.

Allah knows best.