Currently, most doctors and hospitals do not sue their patients that cannot pay, at least here in the Houston, Texas area. They typically send your delinquent bills to collection agencies, that will call and send you letters demanding payment, and put a collection account on your credit reports for 7 years.
One exception here in the Houston area is Memorial Hermann Hospital System, that does file a lot of collection lawsuits against its patients. According to an article in The Houston Press, Memorial Hermann treats patients without insurance, tells them not to worry about the cost and then sues them for thousands of dollars.
But this kind of debt is subject to being discharged in chapter 7 bankruptcy, or repaid in full or partially over time through a chapter 13 bankruptcy. It may be possible to settle it also, if you have no other serious debt problems.