But I think the hospital should still give her treatment though,

Many years ago (in late 80s), I know two brothers, they were Chinese students studied in US (they weren't permanent US resident in that time). They had a bad car accident, one of them broke his leg. They didn't have insurgence, but the hospital let them take the loan to pay back in long term.

I sponsor my two second cousins and their family to immigrant to US couple years ago, so I get to read some documents about emergency healthy care. The documents states that my cousins and their family can not apply any welfare(include medicate) in 10 years, if they do, I need to pay back to the US government. But if they are in emergency healthy condition, they can apply emergency healthy care. I think the difference is that cousins got green cards as soon as they entered US.