Does a med spa in New Hampshire need a medical director?

Short answer
The cited rules do not settle this — none of them require a med spa to name a medical director, but a physician can be disciplined for allowing an unlicensed person to practice in their office, and unlicensed practice of medicine is itself unlawful.

New Hampshire makes the licensee answerable for who works in their office: employing or ALLOWING an unlicensed person to practice there is itself a disciplinary ground. It also treats regular or frequent consultation by an unlicensed person as the practice of medicine without a license, so a consultation exception cannot be used as a standing arrangement. The cosmetology chapter defines esthetics by an enumerated list of surface treatments — giving facials, applying makeup, giving therapeutic skin care treatments, removing superfluous hair, applying eyelashes, beautifying, massaging, cleansing or stimulating the face, neck, arms and shoulders with cosmetic preparations, devices or appliances, and providing pedicure and manicure services — a list on which no procedure that punctures the skin appears.