That's a mischaracterization of what happened. They had several arguments.
1. They believed that she wasn't actually in a persistent vegetative state, but that she was still conscious and could understand what was said to her and display emotions. They even found doctors (who were probably wrong, but they still had validation) to agree with their theory.
2. They believed that her husband wanted her to die so that he could get remarried while still inheriting her estate.
3. They didn't believe her husband when he said that she wouldn’t want to be kept alive via feeding tube. They believed that since she was a devout Roman Catholic she would have seen denying nutrition as euthanasia even though she might not want to be kept alive via a ventilator. This is the main religious argument made at the time, and it’s more of an argument about her wishes than a religious argument.
4. They found doctors who told them that there was a chance of improvement through experimental procedures.
They also seem to have made money since then by taking a salary to run a foundation in her name. It seems like either they were just parents who didn't accept that their daughter was in a persistent vegetative state, they believed the doctors who told them that their was small chance of improvement, they were trying to make money, they really hated their son-in-law, or some combination of all 4.
It didn’t seem to have anything to do with them thinking that God was going miraculously heal their daughter. It’s definitely not evidence of wide scale Christian refusal to accept death.