Praise be to Allaah.
Undoubtedly if this is the case, she has chosen kufr over eemaan. She does not want to stay a Muslim and she is insulting Islam and its symbols, and going against its teachings. In this case she is a kaafir and an apostate, so it is not permissible for him to stay married to her, because Allaah says (interpretation of the meaning):
“Likewise hold not the disbelieving women as wives”
i.e., if he has a kaafir wife, it is not permissible for him to stay married to her. He has to advise her and establish proof against her, then leave her. If he is in a place where there is an Islamic government and sharee’ah law, then he has to refer her case to the Muslim qaadi, for him to ask her to repent. If she does not repent then the ruling of Allaah should be carried out on her, which is execution, because the Prophet (peace and blessings of Allaah be upon him) said: “Whoever changes his religion [leaves Islam], execute him.”
But if that is not possible and there is no Islamic rule or sharee’ah law in his country, then at least he should separate from her completely; it is not permissible for him to live with her after she has clearly stated her kufr.