Is it permissible, Islamically, to join a health insurance plan for childbirth in a private hospital to ensure supervision by a woman during delivery, or is it obligatory to rely on Allah and go to a public hospital?

Permissible insurance is cooperative Takaful insurance, whereas commercial insurance is forbidden. It is permissible for a male doctor to perform a delivery when there is a necessity, but a woman should endeavor to find a female doctor whenever possible. If that is not possible, then there is no blame, due to the Almighty's saying: "And He has already explained to you what He has forbidden to you, except for that to which you are compelled," and because looking and touching for treatment are permissible, even in the private parts, when there is an urgent need, provided there is no woman who can perform it.