I've been in a car wreck and I've received a letter from my own health insurance company demanding that I pay the medical bills back if I get a recovery in my automobile accident case. Do I really have to pay them back and is that legal?

It depends.  In most cases today, insurance policies contain "subrogation" or "reimbursement" provisions that claim that the insurer can force you to repay if you recover damages in a lawsuit or settlement.  While Georgia law generally prohibits an "assignment" of a personal injury case and disallows "subrogation", it does allow for "reimbursement" if you have been made "whole" for all "economic and non-economic" damages.  The burden of proof is on the health or auto insurer to prove that you have been made whole, which is difficult to do if the case results in a general settlement.

If your "insurance" is through your employer, a federal law, known as ERISA, may trump (or "preempt") Georgia law.  If the ERISA plan is "self-funded", you may be stuck having to repay.  In other words, your "health insurance" acts like a glorified loan of funds, not true insurance.