What is the Sharia ruling on a wife giving birth in government hospitals at the hands of male doctors, and is it obligatory to go to an expensive private hospital for her to give birth at the hands of a female doctor?

If the financial costs for treatment in private hospitals are affordable, then one should not resort to male doctors when female doctors are available, in order to safeguard the wife's religious observance and honor. However, if the means for private treatment are not available, and no female doctor can be found, and there is an urgent need for treatment, then there is no harm in going to a trustworthy Muslim male doctor, provided that the husband or a mahram (unmarriageable kin) is present, or someone who would prevent unlawful seclusion. It is a condition for examining for treatment that a mahram or husband be present, and that there is no practitioner of the same gender available. Looking and touching are permissible out of compelling necessity for treatment. And the treatment by a non-Muslim woman is preferred over a Muslim woman if the need is urgent, because her gaze is considered less critical.