May 22 (Reuters) - A U.S. judge on Wednesday allowed a lawsuit filed by Exxon Mobil against two activist groups seeking to bar their climate resolution to go ahead against one of the two groups.

The oil company's lawsuit raised alarm among activists and proxy advisers who argued it would muzzle debate among shareholders and public companies.

Exxon sued climate activist groups Arjuna Capital and Follow This in January and told the court it would not drop the matter after they agreed to withdraw their petition, citing "the likelihood" the pair could file similar resolutions in the future.

U.S. District Judge Mark Pittman on Wednesday ruled that Exxon could continue its case against U.S.-based Arjuna Capital, saying it has authority to hear the case. But he said it could not pursue its claim against Netherlands-based climate activist group Follow This, finding it was outside the court's jurisdiction.

The activist groups argued that Exxon's legal strategy would allow the company to "haul its shareholders into any court in the United States."

The judge also denied Exxon's request for jurisdictional discovery, a pretrial process where a party seeks to obtain evidence to determine whether the court has the authority to hear the case.

Exxon's annual shareholder meeting will be held on May 29. Arjuna Capital and Follow This had asked Exxon to adopt so-called Scope 3 targets to reduce emissions produced by users of its products. Exxon is the only of the five Western oil majors which does not have such targets.

Follow This has made similar proposals at shareholder meetings of various oil majors over the past two years. However, it has been losing shareholder support due to tighter global oil supply, rising energy costs for consumers, and increased energy security concerns following Russia's invasion of Ukraine.

Follow This received approval from 28% of Exxon shareholders who voted in 2022. Last year it received just 10% of the vote.

Exxon shareholders have already rejected Scope 3 targets, the company contends, with shareholders campaigning for changes "calculated to diminish the company's existing business," it said.

Exxon won support from business lobby groups the U.S. Chamber of Commerce and Business Roundtable, which said the case "exemplifies activist groups' takeover of the shareholder proposal process to score ideological points." (Reporting by Sabrina Valle and Gary McWilliams in Houston Editing by Matthew Lewis)