Do you have to pay back health providor after auto settlement?

It depends. Check with your health insurer and ask whether or not their is a subrogation clause in your policy. If there is, then your health insurer will likely want "pay back" of any monies they have paid out on your behalf that are related to the auto accident. If there is no such clause then any settlement money is likely yours for the taking/keeping.

In case you aren't familiar with the term subrogation, most companies refer to subrogation as the process in which expenses are recouped for a claim paid on behalf of the insured when another party should have been responsible for paying some, or all, of that claim.