My dog attacked another dog. I refused to pay vet bills, I thought it would be admission of guilt. What am I responsible for?

Talk with a California attorney. He/she can review the relevant CA statutes with you - in many states, the relevant statutes impose strict liability.

You are liable for any damages caused by your dog. It is unlikely you are liable for the other owner's emotion distress. Don't pay any vet bills without getting a copy of the bill. Speak with a local attorney to draft an agreement limiting your liability to the vet bills as they stand.

You did not state how bad the injuries were to the other dog. If the dog was very badly injured and the owner witnessed the attack, you may be liable for emotional distress damages. Since your dog was not leashed and was off your property (was in the common area rather than contained in the privacy of your apartment walls), you are liable for the damage caused by your dog. If the injuries to the other dog were very minor, then ask for a copy of the vet's bill and draft a simple agreement stating that the payment is in full and final settlement of any injury related to the incident. If the other dog was badly injured, you should consult with an attorney about potential liability.