What happens in America when you are seriously injured in, let's say, a car accident and don't have health insurance?

Hospitals don't just refuse to help, do they?
Hospitals are able to refuse treatment to patients for a variety of reasons, one of which is if the patient does not have insurance [1]. Additionally, hospitals are able to turn away patients in emergencies or critical situations if failing to treat them will result in severe injuries or possibly death [2]. Otherwise, hospitals are required by law to provide patients with the necessary treatment, even if they do not have insurance [3].
Hospitals are required by law to provide treatment to anyone who is injured, regardless of whether or not the individual has health insurance [2]. Even those who are seriously injured can seek immediate medical treatment, as your life is far more important than the costs you will incur [2]. Once your injury has been successfully treated, you can sit down with an administrator at the hospital to discuss your options for paying for your care [3]. In some cases, hospitals will reduce the amount owed dependent on your income, or set you up on a payment plan that will allow you to repay the debt over time [3]. You may also be able to lower the debt if you agree to pay in cash [3]. If you are uncomfortable handling the negotiations yourself, there are a variety of private companies that will handle the back and forth for you [3].