What is the ruling on a pregnant wife going to a government hospital where she knows there are female doctors, but a male doctor might enter or she might deliver at his hands, knowing that her husband objects to that?

It is not permissible for a woman to expose her 'awrah to a non-mahram man unless there is a confirmed necessity, based on the Almighty's saying: "But whoever is forced, neither desiring wrongfully nor transgressing, then there is no sin upon him," and His saying: "So fear Allah as much as you are able." Necessity is realized when a woman needs treatment and there is no skilled female doctor available. Since the hospital has female doctors, it is not permissible for your wife to expose her 'awrah to a male doctor, and you should try to gently persuade her to change her mind.