Birth is not necessarily an emergency room procedure. In order to have the baby at the hospital, they usually require a portion of payment upfront, along with insurance. If there are complications, they might admit you without a prior plan, but you’d be in the emergency room rather than obstetrics.
The second answer is that they will just bill you and you can pay it off over years or just let it go to collections. They have legally more limited ways to collect medical debt. They can harass you and ding your credit but they can’t garnish wages, etc. As long as you are making a minimum payment, they may not be able to send it to collections (sometimes a few dollars a month).