Can I sue the other driver's insurance company in California?

Generally no. California bars third-party claimants from suing the at-fault driver's insurer directly for mishandling a claim (Moradi-Shalal v. Fireman's Fund (1988) 46 Cal.3d 287); the lawsuit runs against the driver. After winning a judgment, Insurance Code section 11580(b)(2) lets the injured party collect from the insurer up to policy limits.

The other driver's insurer owes its duties to its own policyholder, not to you. In Moradi-Shalal v. Fireman's Fund (1988) 46 Cal.3d 287, the California Supreme Court closed the door on third-party bad faith suits: a person injured by someone else's insured cannot sue that insurer directly for unfair claims practices, however badly the claim was handled. The correct defendant is the at-fault driver, under the two-year personal injury deadline of Code of Civil Procedure section 335.1; the insurer defends and pays behind the scenes. The insurer becomes a direct defendant only later: once judgment is entered against its insured, Insurance Code section 11580(b)(2) gives the judgment creditor a direct action to collect within policy limits. Claim mishandling still has consequences: Department of Insurance complaints are available, an unreasonable refusal to settle within limits can expose the insurer to the excess judgment, and your own carrier (on uninsured motorist, collision, or medical payments coverage) owes you good-faith handling, enforceable in a first-party bad faith action.

Authority: Moradi-Shalal v. Fireman's Fund (1988) 46 Cal.3d 287