This is a great question, especially since you live in California where there are so many drivers who may not necessarily own a car. The State of California actually offers auto insurance that is tailored to your exact need in this situation. It is known as SR22 Insurance.
Essentially, this insurance covers the other driver if you are the cause of the accident. It typically consists of bodily injury and property damage coverage for the other driver. However, it is best to ask your insurance agent what the minimum amounts to carry should be as the legally required amount is probably not enough to avoid a lawsuit in the event of an accident.
Make sure to shop around with all of the different companies that offer this type of insurance in California, as the cost of policies can vary widely from company to company. Good luck in your quest for insurance and glad to hear you are being a responsible driver!