I know of several cases where the Doctor treated over the objections of the parents. These were all cases where the child suffered profound and hideous brain damage, with no chance of ever recovering.
It would be interesting to know the hospitals and states. There are some differences.
However, I know that in many religious hospitals there is a member of the senior staff that will be called in if the parents are really stretching out the end, where all hope is gone.
That staff member will quietly sit down with the family and gently explain that extending the process is not going to change the outcome. And that sometimes you have to let go.
I would not want that job.
But, I also know there are people who have overridden the doctors on these occasions.
Most doctors and hospitals will keep going until they are specifically directed otherwise. There are still a lot of folks in the world who think Doctors are God’s hands on earth (most of them Doctors.) If they are not told specifically to stop, in clear words...they will keep going.
I know from personal experience, that Doctors do not take the long term cost of care into consideration at all. That is the adminstrators job.