If you don't have health insurance, check your car insurance policy for "Medical Payments Coverage" (MPC) which might pay for your medical bills after an auto accident. If you don't have MPC, you should call your insurance company and ask about it. It's very inexpensive and very good to have on your auto policy. Even if you have health insurance, they may not pay for everything in which case MPC is a terrific bonus to have! You can't always rely on the responsible party to pay your bills after an accident.
However, keep in mind that if you recover any money from the responsible party's insurance carrier, you may have to reimburse your own insurance company for payments made pursuant to your Medical Payments Coverage. They usually have a right to reimbursement. While this may be the case, there are times when you can negotiate the amount owed, and in some cases, your insurer might agree to a total waiver.