"My wife and I"
Umm, actually, "me" is correct, although it could be improved by: "apologize to my wife and me."
The preposition "to" must be followed by the objective form of "I/me," which is me. One would never say "I want the religious department to apologize to I."
Making the object of the preposition a compound one, by adding "my wife" does not change the fact that the object form of "I/me" must be used.
Thanks.