No, a self-insured car rental company does not have a duty to defend the driver of one of its rental cars. In the aftermath of an accident any legal issues between drivers are just that, legal issues between the drivers. This means that your insurance is the one footing the bill, and you are the one injured parties will seek damages from. In the event damages are rewarded you will be on the hook to pay them. Now there are cases where damaged parties have sought the payment of settlements from self-insured rental companies. But in those cases the legal matter is between the company and the injured party not you. So the only duty a rental company has is the duty to defend itself.
When you rent a car they always offer additional insurance for purchase. This does not mean that the company's insurance policy will cover you and defend you in the instance of a court case. The insurance you are buying is really a waver. It means that the company will not pursue you for damages relating to the car. You will not be on the hook for paying to fix the rental car up. You will be on the hook for everything else. The rental companies service to you terminates with the car. Personal injury is you and you alone.