Sept 30 (Reuters) - California Governor Gavin Newsom on Saturday vetoed a bill that would have paid unemployment benefits to striking workers, and had drawn strong support from labor unions and from his fellow Democrats in the state legislature.

In rejecting the bill, Newsom noted that the state's unemployment trust fund is already nearing $20 billion in debt.

"Now is not the time to increase costs or incur this sizable debt," he wrote in a message explaining his veto.

The Democratic-majority legislature passed the bill in September amid several high-profile strikes. Hollywood writers ended their nearly five-month walkout 12 days later but Hollywood actors remain out on the picket lines. Southern California hotel workers are also on strike.

The bill would have made workers out on strike for at least two weeks eligible for unemployment checks. The vast majority of states, with the exception of New York and New Jersey, do not offer unemployment benefits to striking workers. (Reporting by Joseph Ax; Editing by Edmund Klamann)