Can a non-physician own a med spa in Kentucky?

Short answer
No — for medical services: an unlicensed person may not open or maintain a place of business for the practice of medicine, and the fee-sharing carve-out reaches only licensed practitioners, so a lay owner can't take a share of the professional fees.

Kentucky's licensing prohibition reaches the premises and the holding out, not only the treatment. No person shall engage or attempt to engage in the practice of medicine or osteopathy within the state, or open, maintain, or occupy an office or place of business within the state for engaging in practice, or in any manner announce or express a readiness to engage in practice, unless the person holds a valid and effective license or permit issued by the board. Violating or attempting to violate, directly or indirectly, or assisting in, abetting or conspiring to violate any provision of the medical practice act — including the code of conduct promulgated by the board — is itself a ground for discipline. These claims survey the Medical Practice Act; Kentucky's business-entity and professional-service-corporation statutes are NOT surveyed here, so confirm the entity form with counsel.