Islam has ordered parents to take care of their children and to bring them up according to Islamic manners. The Prophet (peace and blessings be upon him) is reported to have said: “Every child is born on fitra (or innate sound nature) and his parents make him a Jew, a Christian or a fire worshipper.” (Reported by Al-Bukhari and Muslim)
The Muslim father should build in his children good Islamic characters including truthfulness, generosity, love of sacrifice, putting others first, helping others. He should keep them away from evil characteristics. Such good Islamic characters are little different from good Christian characters. In addition, teach the children the Islamic manners in all things: using the toilet, eating, drinking, dressing, sleeping, leaving and entering the house, etc. Try to focus more on those areas in which Christians and Muslims agree, and as the children grow older and can understand teach them more of Islam.
Sheikh Ahmad Kutty, a Senior Lecturer and Islamic Scholar at the Islamic Institute of Toronto, Ontario, Canada, states the following: “As a Muslim father, one must do everything possible within their means to raise the children according to the Islamic faith and practice. Once a father has done his part, he must leave the matter to Allah. He cannot prevent his wife, who is a Christian, from exposing her children to her beliefs so long as she does not force them to adopt them. After they have attained adulthood, it is up to them to choose the religion of their choice; the father is not accountable if they exercise the choice wrongfully. The Qur’an says: “There shall be no compulsion in religion; the truth stands out clearly from falsehood.” (Al-Baqarah: 256)
Allah says elsewhere, “…We have only the duty to convey the message.” (Ash-Shura: 48)
If the children are exposed to points of the Christian theology that directly contradict Islam, you have a religious obligation to clarify the Islamic viewpoints as persuasively as you can, but always exercise rifq or gentleness and hikma or wisdom in teaching your children. It is our firm conviction that all of the essential Islamic beliefs and practices are fully compatible with common sense and reason and can be readily accepted by anyone with sound mind, whereas the man-made theologies are based on intricate, arbitrary rational arguments which can be rejected by the fitra or innate sound nature.
Also a husband in this situation should never stop praying to Allah to guide his wife and children to the true path.