Exxon Mobil Corp. XOM 1.00% , the oil giant long derided by environmentalists, is trying to give itself a green facelift. Chief Executive Darren Woods has called for Exxon to become “part of the solution” on climate change, a point he is expected to make Saturday as he and other oil and gas executives meet with Pope Francis at the Vatican to discuss the issue. The company has flooded the airwaves during the NBA playoffs with advertisements touting its research into fuels made from algae. It pledged last month to cut its methane emissions 15% by 2020. Exxon is now calling for global action to address climate change, and it has begun publicly promoting a U.S. carbon tax. It is part of an evolving shift by Exxon under Mr. Woods, after some of his predecessors adopted a more adversarial stance on climate and environmental issues with governments and activists. […]