Is the husband obliged to give zakat al-fitr on behalf of his Christian wife?

Firstly: 

Zakat al-fitr must be paid on behalf of every free Muslim, male or female, old or young. With regard to the non-Muslim, zakat al-fitr is not obligatory for him, because of the hadeeth of Ibn ‘Umar (may Allah be pleased with him) who said: The Messenger of Allaah (blessings and peace of Allah be upon him) enjoined zakat al-fitr upon the people at the end of Ramadan, a saa‘ of dates or a saa‘ of barley, upon everyone, free or slave, male or female, among the Muslims. Narrated by al-Bukhari (1504) and Muslim (984). 

It says in Mughni al-Muhtaaj (2/122): No zakat al-fitr is required of one who is originally a non-Muslim, because the Prophet (blessings and peace of Allah be upon him) said: “of the Muslims”, and this is the consensus of the Muslims. al-Maawardi said: … because it is a purification, and he (the non-Muslim) is not one of its people [i.e., it is a purification from sin, and there is no purification from sin for the non-Muslim unless he becomes Muslim]. End quote. 

Abu Ishaaq ash-Shiraazi said in al-Muhadhdhab: Nothing is required of him except the zakat al-fitr of a Muslim. If the one on whose behalf he is giving it is a non-Muslim, he is not required to give zakat al-fitr on his behalf, because of the hadeeth of Ibn ‘Umar, “[it was enjoined] upon everyone, male or female, free or slave, among the Muslims”. Moreover, the purpose of zakat al-fitr is purification of the one on whose behalf it is given, and the disbeliever cannot be purified [of sin]. End quote. 

An-Nawawi said: ash-Shaafa‘i and his companions said: He is not obliged to give anything except the zakat al-fitr of a Muslim; if he has a relative or a wife or a slave who is a non-Muslim, he is obliged to spend on them, but he is not obliged to give zakat al-fitr on their behalf. There is no difference of opinion concerning this matter among us. This was also stated by Malik, Ahmad and Abu Thawr. End quote from al-Majmoo‘ (6/74) 

Al-Hijjaawi said in Zaad al-Mustaqni‘: He should pay it on his own behalf and on behalf of any Muslim on whom he spends. 

Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) said in ash-Sharh al-Mumti‘ (6/155): It may be understood from the author’s words (may Allah have mercy on him) that he should not give zakat al-fitr on behalf of a non-Muslim wife or a non-Muslim slave. End quote. 

To sum up: the Muslim husband does not have to pay zakat al-fitr on behalf of his non-Muslim wife. 

And Allah knows best.