Arizona never owned its public lands. It cannot take them back. The US conquered Mexico (nice job, guys) and got the western states for some walking around money for the Mexicans, who were destitute as a result of hideous governments, exempt revenue sources (church and most rich swine) and pizzing away money on the military, which, the joke goes, had more officers than enlisted men.
Arizona did not exist as a state until the 20th century; it was just a territory of the US until then. The US has owned all of ARizona except the parts it sold since the Treaty of Guadalupe Hidalgo and the Gadsden Purchase some years later.
Many of the western states such as Nevada, Utah, Idaho, etc. have huge sections which are the property of the nation. And should remain so. If ARizona was given the Grand Canyon it would be as commercial and ****** up as our Alamo.