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

Short answer
Not directly — no "medical director" by name, but a facility where nurses perform aesthetic procedures must employ a clinical supervisor who is an APRN or other licensed independent practitioner qualified in all procedures offered; an RN or LPN may not serve.

New Mexico wrote a rule for this exact business, and it is recent. 16.12.14 NMAC — "Nursing Practice in Aesthetic Healthcare Facilities", effective 2024-05-21 — defines an aesthetic healthcare facility and requires it to employ a clinical supervisor who is an APRN or other licensed independent practitioner with the training to perform every procedure the facility offers. That supervisor must develop or approve written protocols for all procedures, detailed enough that making independent healthcare decisions never falls to someone without the scope to make them. On the physician side, the medical board treats the use of medical therapeutic and cosmetic devices as the practice of medicine, and a medical assistant may use one only if certified on that device, only if it is non-incisive and non-ablative, and only when the supervising physician is immediately available on the premises.