I was involved in a car accident and was not at fault. I have contacted the other party's insurance company, and it refuses to honor my claim because its insured has not given notice of the accident. Doesn't that insurance company have an obligation to pay my claim even if its insured does not report the accident?

The company is obligated to pay what the insured is legally liable for because of an accident. A ticket by itself is not evidence of 100% legal liability, therefore, in many instances the company may not have to pay for damages if their insured does not report the accident and cooperation in the investigation.