I believe the AMA is responsible for accreditation of medical schools. Without accreditation, they can't make doctors, thus the AMA controls the supply of doctors.
I am not a physician, but I have been in the guts of healthcare for quite a while, and the AMA continues to pop up as the man behind the curtain surprisingly often.
* LCME accredits M.D. Schools
* D.O. Schools have their own accreditation (not sure what it is)
* AAMC runs the racket of "MATCH" to appoint graduating medical students to programs (and strip most of their rights as employees).
* ACGME accredits Residency Programs.