In California, is it legal for a landlord to charge a tenant for liability insurance in addition to rent?

Yes, it's legal. In California, a landlord may require a tenant to purchase a liability insurance policy to cover losses which might occur on the premises through a tenant's negligence, such as property damage (from a fire you carelessly caused) or bodily injury (for example, if a guest is injured through your negligence on the rental property).  Landlords typically require renter's insurance as  a term of  the lease or rental agreement, and set the policy limits or minimum amount of liability insurance the tenant must obtain.

Look over your lease or rental agreement to see whether there is a clause requiring you to pay for liability insurance.  If not, and you have a month-to-month rental agreement, the landlord may be able to amend your agreement (with proper notice) to add a clause requiring liability insurance. If you have a long-term lease that does not include a clause requiring liability insurance, your landlord may need to wait until the end of this lease term to require insurance. Of course, you can agree to purchase the insurance, which may be the best idea. Liability insurance is usually part of a broader renter's insurance policy, which also covers losses from theft, and can be a very prudent investment.

If you're renting in one of the several California cities with rent control, check your local rent control law for information on how and when landlords may require tenants to purchase liability insurance.