WASHINGTON—The Environmental Protection Agency moved to ease Obama-era vehicle emissions standards, siding with car makers who say the rules don’t work in an era of cheap gasoline, and setting up a fight with environmentalists and the state of California.

The rules, finalized in the waning weeks of the Obama administration, would require auto makers to cut emissions enough so that new vehicles sold average more than 50 miles a gallon by 2025. Monday’s decision would start a process to relax future standards covering vehicle...