President-elect Donald Trump nominated Fox News host and combat veteran Pete Hegseth—who has railed against “woke” leadership of the U.S. military—as his Defense Secretary on Tuesday night, marking one of Trump’s most controversial cabinet picks so far as critics raise concerns about his lack of experience and right-wing views.

The choice of Mr. Hegseth, a dedicated supporter of the president-elect in his first term, is outside the norm of the traditional choice for the post.