The infant is also financially liable but likely uncollectable. The baby's life may or may not be ruined depending on where the US goes w/r/t health insurance coverage for pre existing conditions.
In the US only the wealthy ( millionaires ) are safe from medical bankruptcy and even they can get taken down by some issues.
I mean, can you even imagine other areas of life where every person you purchase from attempts to screw you over by demanding 2x-10x of the reasonable fee unless you had the information to be aware of it? It would be impossible, because everyone would just take their business somewhere else.
But you can't exactly say "oh, I'm not willing to pay my life savings for this, I'll just take this 3 months prematurely born baby to the shop across the street, or maybe get it done in six months time instead, when it's more convenient.".
As usual, this is the craziest part of the comment to my European mind. Your healthcare system spent somewhere around an entire lifetime of effort for something that, while relatively dramatic, should still be quite routine for a sophisticated medical organization!