If the woman is an adult, the family can simply tell them to go fuck themselves. There is no such thing as debt inheritance with the exception of with spouses because they are "one estate".
If she had health insurance, the insurance company should sue the state (as well), because they can better afford to. And because this won't be the only time.
In a catastrophic case like this, they wind up spending more than the family, even though it's a smaller percentage of their assets. So they have every incentive, and I'm sure the family would be willing to testify towards their case.
Schiavo’s parents wanted to keep her on life support; her husband didn’t. I wonder if he was expected to pay too.
I imagine hospitals have all kinds of ways of getting people to sign some agreement to be held responsible for all kinds of costs. The entire system is fucked - you don’t know how much you owe until you start getting random bills months later.
In this case, the hospital doesn't have to do anything like that because the anti abortion law itself is coercion enough. If the family does anything that could even potentially contribute to the fetus dying, such as refusing care, they'll be prosecuted for murder. That they can't afford the care is immaterial.