May 23 (Reuters) - The California State Teachers' Retirement System has voted against re-electing Exxon Chairman Darren Woods and Lead Director Joseph Hooley, joining other public systems concerned the energy company's lawsuit against climate activists could diminish investor rights.

In a statement sent by a representative, CalSTRS said the votes were "a measured approach to hold two major leaders of Exxon accountable to investors." An online record shows CalSTRS cast votes for the company's 10 other director nominees.

The votes by CalSTRS for Exxon's May 29 annual meeting were in line with previous comments by one of its board members, State Treasurer Fiona Ma, expressing concern about the suit. CalSTRS still took a softer approach than the state's other big pension system, for public employees, which said it would vote against the whole board.

Neither system is among Exxon's 40 largest investors, but they have played influential roles in previous Exxon director elections.

In January Exxon sued activists to block a vote on a climate proposal they submitted, sidestepping the usual regulatory process. Although the investors withdrew their resolution, Exxon continued the lawsuit, seeking legal costs and other relief.

On Thursday, a U.S. judge allowed Exxon's suit to proceed against one of the two activist groups it had sued, Massachusetts-based Arjuna Capital. The judge also said Exxon could not pursue its claim against Netherlands-based group Follow This, as it was outside the court's jurisdiction.

Exxon said in a statement the ruling moved it closer to its goal of reforming the shareholder resolution process. It has previously said securities regulators have allowed too many resolutions to come to a vote, costing companies money.

"It's time to stop the abuse of the system, and we’re pleased the judge agreed that we’re entitled to our day in court," Exxon said.

Asked about the CalSTRS votes, an Exxon representative reiterated its past comments its board has overseen significant shareholder value creation.

(Reporting by Ross Kerber. Additional reporting by Sabrina Valle)