I am not sure if your car insurance is the first payer or your health insurance is.
If you have an outpatient surgery, your co-insurance shouldn't cost you too much (I suppose 20%?); if it's inpatient surgery, you probably will reach your out of packet maximum (check your policy see how much it is). After you reach your OOPM, your health insurance will cover 100% of your medical bill.
I believe there is a max if your car insurance wants to raise your premium.