Biden also nominated Jack Markell, the former Democratic governor of his home state of Delaware, to be the U.S. representative to the Organization for Economic Cooperation and Development, the White House statement said.
The pair were among 17 nominations sent to the Senate for confirmation.
McCain endorsed Biden in November’s election in which the Democrat defeated Republican President Donald Trump, who had repeatedly criticized McCain’s husband. John McCain died in 2018.
McCain is chair and director of the Phoenix-based Hensley Beverage Co, which distributes wine, spirits and non-alcoholic beverages.
Other nominees included Massachusetts Democratic state Representative Claire Cronin as U.S. ambassador to Ireland and Fiona Whelan Prine, widow of the late singer-songwriter John Prime, to the National Council on the Arts.