Your concern in very valid. Coulterville is quite a drive from John C. Fremont Hospital in Mariposa. To avoid the winding drive over the Bagby Grade, most people in that area opt for Tuolumne County medical services - in Sonora. Neither are close by. If you have medical concerns, I would recommend staying closer to Mariposa or Sonora.
John C. Fremont Hospital is in Mariposa, with Mercy Medical providing ambulance service. There are doctors offices also in Mariposa. For more extensive medical services, you would go to Modesto, Merced, or Sonora.