The only guidance the U.S. Constitution gives on filling Congressional seats is that they must be filled through elections. How seats are filled in the interim if someone resigns, is too ill to serve or dies, is up to each state.
Arizona state law requires that the governor of the state appoint someone to fill a vacant U.S. Senate or U.S. House seat. In the case of a Senate seat, the governor, currently it is Republican Doug Ducey, appoints an interim senator. That person, who by state law must be of the same party as the person who resigned or died, would serve through the next scheduled statewide general election. That is Nov. 6 in this case.
Whoever wins the election in November would serve the remainder of the term. In this case, McCain's term goes through 2022.
A spokesman for Ducey told The Arizona Republic that the governor will wait to name McCain's successor until after the senator's burial at the U.S. Naval Academy Cemetery in Maryland.
"Out of respect for the life and legacy of Senator John McCain and his family, Governor Ducey will not be making any announcements about an appointment until after the Senator is laid to rest," Ducey's senior adviser Daniel Ruiz II said in a statement. "Now is a time for remembering and honoring a consequential life well lived."
In McCain's case, however, things may work a bit differently. Arizona law states that the seat must be filled by an election, but the last date to file as a candidate to be on the ballot for the Aug. 28 primary election was May 30 at 5 p.m.
According to a post from the Tucson Sentinel, Arizona courts have ruled that an election is to be held on Election Day (Nov. 6 this year). There have been no rulings on if that applies to an election to fill a vacant Senate seat.