On our post, we have cdc centers and fcc homes all run through the Army. There are full day options or you can do hourly care for $4 an hour. If your husband is deployed, then you get an additional 16 hours of free care. The centers and homes run through the Army are held to very high standards. FCC also has homes that are off post as well. In our area there is also a reputable private center off post that offers hourly care. Your child has to be signed up with CYS to use the cdc or fcc childcare options, which can take a little bit of time to get through, but it is worth it in the long run.
I do not know how old your child is, but I have a 2 year old and he usually goes with me to appointments if they are at the hospital on post. They have no problems with this, as long as he can be restrained in a stroller while I am having an exam. If you need x-rays, you can always just wheel the stroller to the safe area where a parent would stand when their child gets an x-ray. If I have an appointment off post, like a dental checkup, I either have my husband take off s couple of hours (they are really good about helping the families out here) or I take my on to the cdc or fcc.
The gym here has a small cdc so you can workout in peace, and I think it is a great idea to have one in the hospital as well!
Added- Beenthere: The hospital that I go to is not just for when you are sick. The hospital has TONS of clinics with specialized doctors. If you are going in for OB/GYN, or the foot doctor, or to get the yearly asthma test done, I don't see the problem with it. Maybe there could be a way for the parents to be limited to using the center only if there are not ill.