All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad, sallallaahu ‘alayhi wa sallam, is His slave and messenger.
You should know that being a Muslim is a prerequisite for having the custody of a child according to the preponderant opinion of Muslim jurists. Therefore, it is impermissible for you to approve of leaving your daughter to be raised by her non-Muslim mother. It is incumbent on you to strive to take her in your custody. If you tried to accomplish that objective but failed, you are not blamed for that. Allaah, The Exalted, says (what means): {Allaah does not charge a soul except [with that within] its capacity…} [Quran 2:286]
In this case, you should take good care of your daughter, advise and direct her to the right path, and keenly instill in her mind and heart the Islamic beliefs, morals, and manners.
On the other hand, it is impermissible for you to receive training and acquire new skills in a place where impermissible intermixing between the two sexes is required as clarified in Fataawa 87469 and 90355.
As for touching the patients during the medical training, even if this entails touching the patient’s private parts, it is permissible as long as the due conditions are observed. Please refer to Fatwa 127985 in this regard.
In brief, this matter should be addressed in accordance with necessity. If it is available for the male trainees to examine male patients, it is impermissible for them, then, to examine female patients during the medical training.
Our advice to you is not to leave the medical field or your specialty; instead, you are advised to find a Muslim country where you can go through this medical training without having to commit all these religious violations in order to maintain your religiosity. Verily, the Muslim’s religion is the most precious thing he can ever possess. If he loses his religion, he has lost everything. It should be noted, though, that staying in a Muslim country will not prevent you from taking care of your daughter and monitoring her upbringing. The means of communications have developed and communication has become easy nowadays.
Allaah Knows best.