In a California showdown over corporate governance, the largest U.S. public pension fund is attacking Apple Inc. for the way it handles board elections.

Apple's directors can currently hang on to their seats with a single "yes" vote in uncontested elections. The California Public Employees' Retirement System wants Apple and other U.S. companies it invests in to adopt rules requiring directors to win a majority of the vote, saying that will make board members more accountable to shareholders.

Calpers...