Kelli Ward, the chair of the Arizona Republican Party, told Mr. Ducey on Twitter last week to #STHU — the hashtag for “shut the hell up” — after he defended the state’s election process. At a news conference, Mr. Ducey responded, “I think what I would say is the feeling’s mutual to her, and practice what you preach.”

Separately, Representative Andy Biggs, a Trump loyalist, singled out Mr. Ducey for a public rebuke over the coronavirus, theorizing that the governor “intends to coerce vaccinations.”

Mr. Ducey’s chief of staff, Daniel Scarpinato, then entered the fray, tweeting to Mr. Biggs: “We always knew you were nuts, but you’ve now officially confirmed it for the whole world to see. Congratulations. Enjoy your time as a permanent resident of Crazytown.”