Is it permissible for a male student in medical school to shake hands with one of his female friends if she initiates it, taking into consideration the nature of his future profession, which will entrust him with the private parts of women?

It is not permissible to shake hands with a non-mahram woman, even if she initiates it. The Prophet, peace and blessings be upon him, refrained from shaking hands with a woman who extended her hand to him, and he is the best role model. It is not correct to use the permissibility of uncovering one's awrah for medical treatment as evidence for the permissibility of shaking hands, because uncovering the awrah is only permissible out of necessity, whereas shaking hands is originally forbidden.