WASHINGTON—Sen. John McCain’s death touches off a number of questions about who will succeed him, putting the spotlight on Arizona’s Republican governor, Doug Ducey, who will choose a replacement.

A spokesman for Gov. Ducey said Sunday any appointment won’t be announced until after Mr. McCain has been laid to rest. “Now is a time for remembering and honoring a consequential life well lived,” the spokesman said.

The governor, who is up for re-election this year and expected to face a stiff general election challenge, is under intense pressure from rival wings of the Republican Party, including factions that want him to appoint someone loyal to President Trump.

The question of succession practically has been a forbidden subject in Washington, as Republicans had sought to show respect for one of the party’s few elder statesmen while he battled an aggressive form of brain cancer.

Mr. McCain was re-elected to his sixth term in 2016. His successor would serve for roughly the next two years, with voters choosing a new senator in the 2020 elections.