It will be up to state laws and the terms of your girlfriend's insurance policy whether they can deny this claim for the damages sustained in the accident with you driving the car.

If her policy requires the driver operating the car to have a valid license, which you did not have, then they can deny the claim. Also it does not sound as if you were listed on her policy, since you did not have a valid license, so they may refuse to cover the accident over this issue as well.

If her insurance company is denying the claim then both the towing and storage fees and the damages to her vehicle would not be paid for by her insurance carrier. Your girlfriend can read through the terms of her policy and ask her agent why specifically the claim is being denied and if she wants consumer help with the situation she can contact your state's insurance regulator.