There are quite a few clinics on the island (speak english) that can take care of all small injuries and make many different tests. You will find them in the main villages (Sairee, Mae Haad and Chalok).
Koh Tao International Clinic (a part of Bangkok Hospital) has english speaking doctors and staff. Private physician clinic and pharmacy with english speaking doctors and staff. In-house care, accident and emergency services. Medical consultation. All major credit cards and travel insurance accepted. Qualified doctor and nurse on site. Overnight admissions. 24 hour emergency service.
Tel. +66 (0) 77 456 037, +66 (0) 81 737 55 44, 9/32 Moo 1 Sairee village.