What is the ruling on a son placing his father—who is in need of intensive care—in a nursing home, or bringing him to his own private home, given that his wife refuses that? Is she permitted to refuse in the first place?

It is the wife's right to reside in an independent dwelling with its own facilities. If you house your father with you and provide your wife with an independent section of the dwelling, then she has no right to object. If this is not possible, you can reside near your father to attend to his affairs, or hire someone to care for him. The spouses should reach an understanding, and it should be clarified that the wife's kindness to her husband's family and her assistance in helping him honor his parents are part of good companionship.