The match is binding upon both parties. However, the match commitment is worded thus:
So, it is quite possible (but rare) to match into a program where you do not meet their employment policies. An example of this would be failing a drug test -- if you're told that being employed by a program involves passing a drug test, you match there, and then fail the test, you do not get hired (and it's not a match violation for the program.
Almost everyone's employment policies include a statement like: "You must have all licenses/certifications, be physically present, and be medically fit to perform the job starting on xxx". So, failing the boards or not getting licensed would all be reasons to negate a match.
The "Medically fit" is the tough one. I assume that's what happened here -- they sent you a letter saying you were not medically fit. You can try to fight this -- either with internal reviews or in court -- if you think it is unfair.
Programs will certainly want to know the details, mainly looking to see if the issue is resolved (or was made into a bigger deal than it really is). You can simply state that "I was considered medically unfit to start work, but that issue has now resolved and will no longer be a problem" but I expect many PD's will not be interested, worried that you might not be telling the whole truth.