There are some open questions left by your post - it would be helpful for you to clarify. Do you contest that you were at fault? Who does the second insurance company cover, you or another driver? What happened in the accident? Did the police do a report and make a finding? Do you have your own policy for a car that you own? If you do, you need to tell them about the accident and ask them to defend you because yes, the owner can sue you for the deductible he had to cover.
The short answer is yes. The longer answer is that depending on the situation, if he sues you, it may be his own company that is defending and indemnifying you because you had permission to drive his car. The question here is really whether your own insurance, or his insurance, will cover it. California insurance regulation determines which insurance is primary, yours or his.
This is general advice and does not establish an attorney-client relationship.
First his insurance should cover you the permissive driver, just as they would cover the owner. If you are found at fault they should pay for the other vehicles damages. His vehicle should be covered under the collision coverage. However, it will seem reasonable if you were at fault you should pay the deductible. If the other vehicle is found at fault they should pay everything.