Mexican Gray Wolf Recovery Plan Act of 2016 This bill: requires the U.S. Fish and Wildlife Service (USFWS) to publish a revised recovery plan for the Mexican gray wolf populations in Arizona and New Mexico; outlines what must be contained in the plan, including input from states and individuals, a maximum population of the wolf, and a specified range for the wolf; establishes a process for the state wildlife authority of Arizona or New Mexico to assume or supplant the USFWS' authority to manage such wolf in the relevant states if certain conditions are met; requires USFWS to reduce the population of such wolf within the specified range when the wolf's population exceeds the maximum population; and sets forth requirements for removing such wolf from the list of endangered species, threatened species, or experimental populations under the Endangered Species Act of 1973 if the population recovery goal outlined in the plan is met.