Lawyers for Exxon Mobil Corp. stood before a New York judge in August and told the state’s attorney general to “put up or shut up” after spending three years investigating the company’s public disclosures about climate change, saying authorities should sue the energy giant or move on.

On Oct. 24, New York chose to strike.

Attorney General Barbara Underwood filed a fraud lawsuit against Exxon in state court in Manhattan, accusing the company of misleading investors about how future regulations...