I have no personal experience with serious medical emergencies in Belize, but I think Diane is right (it is becoming more and more true that living in the US does not mean you have access to health care), but with some caveats.
It depends on where in Belize you will be living. If you will be living on Ambergris Caye, I think most kinds of health care is easily accessible, no more than a short plane ride away and local doctors are available for primary care (like well-baby care).
If, on the other hand, you are a missionary going to a remote posting in the Toledo District, there is virtually no medical care beyond the very rudimentary, if that. You are a few hours away from most health care.
If you are a young healthy family with the typical well-baby care needs, my understanding is that outside of such remote areas, basic primary care is very accessible and good and you should have no medical concerns.
However, if you will be living in remote areas in Belize (not AC) you need to educate yourself about tropical diseases and how to protect your family, as infants are very susceptible. (Don't ask MD's in the US, they don't have a clue...get on the net and do your own research about things like home spraying of DDT and stuff like that (www.malaria.org
is a good place to start).
Belize (from what my friends tell me) lacks sophisticated medical practitioners. For example, I have a friend with a serious endocrine problem and she has had to travel outside the country for treatment because the doctors in Belize do not have the knowledge and experience to figure out what is wrong with her and how to treat it.
On the other hand, you could, as Diane points out, have the same problem in many rural areas of the US.
Also, there are many excellent specialists in Mexico. From what I have heard, medical care for middle class people in Mexico is quite good (in major urban centers), comparable with US cities and at much lower costs. One of the reasons is that Mexican doctors are allowed to be doctors and not HMO bean counters.
One thing about health care in Belize, Mexico and many places outside the US that can be deceptive is that less money is spent on fancy offices and accoutrements. Excellent doctors may have small, plain, very basic offices without all the fancy advertising stuff that you see in the US. Hospitals are smaller and less fancy. So long as they are clean, it does not mean that they are not good. It does mean that care costs less.