If a doctor forged your signature in bad faith, that is a very serious violation of law. If a doctor were to do that in order to defraud an insurance company it would be a crime. If it was done to make a fake medicare bill, then it is a federal crime. It is also something that could lead to license discipline issues for the doctor or hospital staff.
However, this issue comes up often. In analyzing this problem, I would focus on the harm to you. If you were harmed, and can put a value on that harm, you can sue for damages in small claims or civil court.
If you do not have a harm you can put a value on, but you are angry and want something done, you can: 1) complain to the licensing board in your state (I'm not sure what kind of doctor this was); 2) complain to the hospital; and 3) (if you think this was an insurance fraud problem, which sometimes does happen) contact the local district attorneys office (which probably has an insurance fraud division).
If you do want to recover money, you can sue for all sorts of causes of action such as performing procedures without your consent, violation of privacy, breach of contact, fraud, etc.
In small claims court, you can do it your self, without an attorney. In the larger civil courts, you would need to hire a private attorney to bring the claim effectively.
Should you settle?
Once you make a proper claim, based upon real evidence, settlement is always a good option. If they are willing to pay a valid claim, and you are willing to accept the offer, yes, settle.
The problem is this: what is your claim worth? What have you lost, and what is the value of that? What can you convince the insurance company it is worth? What damages can you prove up at trial if they refuse to settle with you?
These are all questions that must be answered when evaluating a settlement offer.
In a claim like this, an insurance company would defend the claim. Then you would have to deal with their attorneys.