Protecting America's Wilderness and Public Lands Act This bill provides for the preservation, conservation, and recreational use of public lands, including in Arizona, California, Maine, North Carolina, Oregon, Virginia, Washington, and the U.S. Virgin Islands. In Colorado, the bill designates wilderness, potential wilderness, a recreation management area, wildlife conservation areas, a national historic landscape, and special management areas and establishes the Greater Thompson Divide Fugitive Coal Mine Methane Use Pilot Program, the Fugitive Methane Emission Leasing Program, and a recreation area. In California, the bill establishes a recreation area, partnerships, and a special management area; designates a trail, wilderness, potential wilderness, and wild, scenic, and recreational rivers; establishes a special conservation management area; establishes the San Gabriel National Recreation Area Public Advisory Council and the San Gabriel National Recreation Area Partnership; and adds specified areas to the Rosie the Riveter/World War II Home Front National Historical Park. In Washington, the bill designates wilderness and wild, scenic, and recreational rivers. In Arizona and Oregon, the bill withdraws specified federal lands. In North Carolina and Virginia, the bill requires a study of designating an area as the Great Dismal Swamp National Heritage Area. In Maine, the bill designates a recreational river. In the U.S. Virgin Islands, the bill establishes a heritage area. In Virginia, the bill adds certain lands to Rough Mountain and Rich Hole Wildernesses. The bill extends the Cape Cod National Seashore Advisory Commission until September 26, 2028. The bill establishes an outdoor recreation legacy partnership grant program. The bill establishes a National Heritage Area System.