This depends on the state where the car accident happens and the insurance policies of those involved. For example, some states have a "no-fault" policy, which means that auto insurance will pay a certain amount of your medical bills no matter who was at fault in the accident.
Texas, however, is not a "no-fault" state, so if you get in an accident in Texas, you'd most likely be responsible for paying your own bills. Sometimes there's an exception to this rule. If the other party involved has an insurance policy that has "med pay" your bills may be covered up to a certain amount, which is usually around $10,000.
Having a policy with "med pay" is not required by the state, so it may not be very likely that the vehicle at fault for the car accident has this policy.