You could do philosophy and lean into bioethics (though this might be getting a bit more applied than what you want). There are some great pieces out there on the construct of medicine as a profession, exploring what it means to care for someone and to be responsible for their body. There's also the philosophy of medicine itself, which is a little bit more based in the epistemological commitments of medical diagnosis and clinician judgment.
Narrative medicine (which might be closest to what you're looking for) explores the patient and the physician in terms of their lifestory and their truth. Check out works by Rita Charon and her amazing group at Columbia. Several top medical journals publish narratives and poems these days (the JAMA group comes to mind). I know several undergrads have started literary journals in narrative medicine for pre-meds to explore this perspective early on.
There's been a big push in medical schools to include visual arts in the curriculum as a way to exercise observation and empathy. So you could address the art of medicine. I did a seminar in undergrad that was on Narrative, Theater, and Medicine, in which we studied passages from Shakespeare, Tolstoy, and Chekhov to understand the human condition. These were all really fascinating ways to use the humanities to use the same faculties of empathy that doctors must employ everyday.
I think end-of-life issues provide a wonderful tie-in for medicine and the humanities. It's extremely difficult to discuss what you what at the end of life, what you want your legacy to be. The arts provide a nice proxy by which people can communicate and reflect on what they want. I've organized some workshops that incorporate film, dance, and theater to begin conversations about advance care planning. There's been an onslaught of films (both fictional and documentary) that have addressed these.
Several medical schools have distinction tracks in bioethics and medical humanities, so you can definitely carry on research and activities in this area. I concur with the above poster to check out ASBH, as well as The Hastings Center, Kennedy Institute of Ethics (Georgetown), and Berman Institute of Bioethics (Johns Hopkins). The last one has an excellent weekly email that compiles bioethics/philosophy of medicine in academic literature. It's a great starting point.
A few things to check out:
Wit (film or play)
The Death of Ivan Ilyich by Leo Tolstoy
The House of God by Samuel Shem
poems by William Carlos Williams and Rafael Campos