If you need emergency care, you don't have to time to worry about how your bills are going to be paid. Frankly, automobile liability insurance carriers that represent the other driver in an accident do not pay your medical bills as they are incurred. They only pay you to reimburse you for all your bills at the time of settling your case or, if you cannot settle your case, after you win your case at trial. Most medical providers don't want to wait months or even years to get paid and may pursue collection from you while your auto accident case is pending.
Therefore, you have two options when it comes to giving the hospital your insurance information. If you have "medical payments coverage" on your own auto insurance policy, you can give the hospital your own auto insurance information and they will bill your auto insurance under that coverage first. If you are not sure whether you have medical payments coverage on your auto insurance, or if you know you don't have such coverage, then give the hospital your own health insurance card. These two types of coverage---medical payments coverage and health insurance coverage---will pay your bills immediately so that you don't have to worry about dealing with bill collectors while you are getting treatment for your injuries. Hopefully, you will get enough in your settlement or jury verdict to reimburse you for all of your medical expenses. You may have an obligation to reimburse your health insurer if you do get a settlement from the other driver's auto liability insurer, so make sure you examine your health policy thoroughly to see if this is the case. If you do see an obligation in your health insurance policy to reimburse the health insurer when your accident case is resolved, it is a good idea to contact an attorney to help you understand the actual rights to reimbursement held by your health insurer.