Can a driver with no valid driver's license sue you in an accident that wasn't your fault?

Yes you can be sued for an accident. The requirements are not a license or even if the other party has valid insurance. Just that someone is liable (at fault), there are injuries, the at fault party caused them and the cost to fix them. I understand you believe that the accident was not your fault. Assuming that is correct, the other party can still sue you, and YOUR insurance will defend you fully. But, if you are found to be at fault, or liable for the accident then the third party has a right to collect for his economic injuries only. Under California law, if a party WHO IS NOT AT FAULT, is uninsured, or under the influence they are entitled to be compensated for their economic damages only (medical bills, lost wages, etc.), and cannot collect for Non-economic damages (pain and suffering).