Skechers U.S.A. Inc., TiVo Corp., and other California-based companies that have only men on their boards would need to add at least one woman under a bill being considered in the state’s legislature.

California would be the first state in the nation with such a requirement for publicly held companies, if the state Senate proposal passes. So far, it’s been approved 5-1 by one committee and is headed to another committee April 24 before it can go to a vote...