Does a med spa in Ohio need a medical director?

Short answer
Not directly — Ohio imposes no med spa license and names no "medical director," but applying a light-based device is itself the practice of medicine, so a non-physician may only do it as the delegate of a supervising physician.

Across the acts these chapters cover — light-based devices and the administration of drugs — Ohio puts a delegating physician behind the act, answerable to the Board for it, and the esthetics license itself authorizes nothing that breaks the skin. Which rules that physician is held to turns on the act, not the job title: an injection delegated to an unlicensed person runs through chapter 4731-23, and light-based devices split into four separate lanes that disagree with each other. Ablative may never be delegated at all; a vascular laser may go to a nurse under an on-site physician watching at most two people; hair removal may go to a certified non-licensee under a physician watching up to five, who need not always be in the building.