Is it permissible for me to take medication to induce menstruation on my own, given its delay and irregularity, and my refusal to be examined by a non-Muslim male doctor?

It is permissible to use medicine if it does not result in health harm, and specialized doctors must be consulted. If a woman is compelled to see a doctor and cannot find a female doctor, she is permitted to consult a male doctor, while adhering to the Sharia guidelines, such as avoiding privacy (khalwa) and not touching or looking at what is not necessary.