Currently, Cottonwood Medical Center specializes in Family Medicine, Internal Medicine/Pediatrics, Internal Medicine and Nurse Practitioner with 7 physicians. Cottonwood Medical Center Office Locations. Cottonwood Medical Center Ltd. 560 N Camino Mercado Ste 7 Casa Grande, AZ 85122 (520) 836-5538. Monday : 8:00 AM - 5:00 PM:.