Can a non-physician own a med spa in South Dakota?

Short answer
Not if the entity is a medical corporation — every shareholder, officer and director of a corporation formed to diagnose and treat patients must be licensed under the Medical Practice Act; the cited rules don't cover esthetics-only salons.

SDCL §47-11-3 closes the obvious workaround, and it is the sentence to read first. All of the officers, directors, and shareholders of a medical corporation shall AT ALL TIMES be persons licensed pursuant to the Medical Practice Act. No person who is not so licensed shall have any part in the ownership or control of such corporation — and no proxy to vote any shares may be given to a person who is not so licensed, which closes the route of leaving title with a physician while moving the votes elsewhere. In a corporation formed under the chapter, one or more persons may act as the SOLE stockholders, directors or officers. And persons licensed under the Medical Practice Act, corporations formed under chapter 47-11, or any combination of them, may form a limited liability company to own, operate and maintain an establishment for the study, diagnosis and treatment of human ailments and injuries. Chapter 36-4 then names the arrangement itself among the acts that are unprofessional or dishonorable conduct: Splitting fees, or giving to any person furnishing a patient any portion of the fees received from the patient, or paying or giving any person consideration of any kind for furnishing a patient — and, separately, employing what is known as cappers or steerers.