A hospital itself usually does not sue the patients for unpaid medical bills.
Instead what usually happens when a patient of a hospital has unpaid hospital bills is they send the medical bills off to collections after a year or so of not paying the medical bills.
Then a debt collector will start harassing you through telephone, mail etc and if the debt is large enough then the debt collector could take you to court and sue you for the money owed to them.
The debt collector has to prove though that you owe them the money so they better be prepared to show the proof.
They might win the judgement against you and garnish your wages, levy bank accounts and could even take your home or make you sell your home or other property to pay back the medical bills owed.
If possible work something out with the hospital and pay what you can per month to help get the medical bill debt paid down.
Because having unpaid medical bills can also affect your credit score as well and will stay on your credit report for at least 7 years.