Does a med spa in Idaho need a medical director?

Short answer
The cited rules do not settle this — Idaho defines a "supervising physician" as the person responsible for directing and supervising medical personnel and the patient services they provide, but they impose no medical director requirement on a med spa itself.

Idaho defines the supervising physician as reaching unlicensed personnel, and that is the provision a med spa turns on. a “supervising physician” is a physician responsible for the direction and supervision of the activities of, and patient services provided by, licensed or unlicensed medical personnel — so the statute contemplates unlicensed staff working under a physician and puts the physician on the hook for what they do. Failing to supervise the activities of interns, residents, nurse practitioners, certified nurse-midwives, clinical nurse specialists or physician assistants is a ground for medical discipline, and so is knowingly aiding or abetting any person to practice medicine who is not authorized to. The practice of medicine itself is defined broadly, reaching investigation, diagnosis, treatment, correction, prevention of, or prescription for any human condition by any means or instrumentality involving the application of principles or techniques of medical science — and, separately, offering or holding oneself out as able to do those acts.