Nothing at all will happen if you do not pay medical bills; my brother-in-law had a liver transplant back in the day and they owed a zillion dollars but eventually the hospital, etc. writes it off (after about the length of time it may take you to get SSI :-)

However, it does influence any future credit, and private practitioners might not choose to work with you, etc., but no one is going to haul you into court to get a pound of flesh or anything.