What is the ruling of Islamic law regarding a man resorting to a male doctor to perform a sensitive operation on his wife, due to the unavailability of a specialized female doctor, and what is the expiation for the sin for whoever does that?

The matter concerning the order of treatment for a woman, when it necessitates exposure of her private parts, should ideally be with sufficiently experienced female doctors. Otherwise, it is permissible to move to the next in order of preference, as dictated by necessity or need. There is no blame upon you for what you have done, as long as the situation is as described regarding necessity or need. Therefore, relieve yourself of this burden and do not destroy yourself with worry, sadness, and the whispers of Satan. What you did by presenting your wife to this male doctor out of necessity does not contradict your zeal for your religion and your honor.