President-elect Donald Trump’s pick for secretary of defense, Pete Hegseth, is already facing mounting criticism and hasn’t even begun his duties.

The Associated Press published an article accusing Hegseth of being a white supremacist after revealing that the former Fox News host was removed from National Guard duty during President Joe Biden’s 2021 inauguration because of a tattoo on his arm. The tattoo is a Latin phrase that reads “Deus Vult,” which means “God Wills It,”...