We're missing information. They probably didn't just say "there was an issue" with your policy. They also told you what the issue was. But you didn't tell us what the issue was.
Of course the guy can get his own car fixed without your permission. It's his car! He made his claim to GEICO, his own insurer, and they paid it. Why would you be involved with that decision? He was insured with GEICO and he made a claim with his insurer. If you have no coverage for this incident, the insurer can come against you personally for the funds it has paid to fix the damage.
Not legal advice as I don't hold California licensure. It's just my analysis of the facts you present in light of general principles of law. If you need legal advice, please consult a lawyer who is licensed in California. Lord knows there are hundreds of thousands of them.
I think the first answer on this crash & insurance question is 100% correct. If you are unhappy with the clarification of the problem with your policy given by GICO, file a complaint with the California insurance commissioner, or talk to an insurance coverage lawyer.
Yes they can and should pay the damages. Why would you not want to pay damages to the fellow? If the issue is that you were uninsured then you will be paying Geico back.