Is learning medicine considered a defiance of the mother's wish, given that she threatened to withhold her prayers from her son if he studies medicine?

If the mother's refusal for her daughter to study medicine is due to شرعية [Shar'i] corruptions or harm that may befall her, then obedience to the mother is obligatory. However, if the prohibition is out of stubbornness without a valid reason, then the daughter is not obligated to obey her mother in abandoning the study of medicine, and she will not be considered disobedient for doing so.

A person is obligated to obey their parents in matters that do not involve disobedience (to Allah), and in matters that benefit them (the parents) and cause no harm (to the child). Parents are not allowed to compel their child to do something that harms them or that they do not desire. If the child's maturity and sound judgment are established, the father does not have the right to prevent them from something that benefits them, whether in their religion or their worldly life. And she must be dutiful to her mother, treat her kindly, and not mistreat her.