Can an insurance company claim damages from a person who was involved in an accident with the insured after his vehicle is repaired by the insurance company?

Yes; the process is called subrogation. The insurer that paid for the repair of its's insured's vehicle succeeds to the right of action against the at-fault party for the purpose of collecting that which it paid. The subrogating insurer has no greater rights than its insured did, such that if its insured is found to have been, for example, 30% At Fault for the collision (in those states that adhere to comparative negligence), the insurer can recover only 70% of its damages.