employee's rights after getting into car accident with company car

This question would be easier to answer if we knew who pays for the insurance and who is the named insured on the auto policy. My sense is that your employer is the named insured, because he is the one asking you to pay, not the insurance company.

The simple answer is that no, your employer cannot compel you to pay for something that you are not under contract to provide. The more complicated one is, was there some kind of agreement between you and your employer to provide for things like this when the car was provided for you. I doubt it, so my former answer stands.

While he can't force you to pay, remember that Washington is an "at-will" employment state, which means your job can be terminated at will. If you like your job (despite the unreasonable requests by your boss), consider talking to a supervisor. If there's no one above him, you might offer to pay a portion of it with the understanding that you had no notice of the obligation when you accepted the use of the company car.

Good luck.