The Florida governor’s race is shaping up to be a ferocious showdown after Tuesday night’s primary victories by Representative Ron DeSantis, the Republican endorsed by President Trump, and Mayor Andrew Gillum of Tallahassee, a progressive Democrat endorsed by Senator Bernie Sanders of Vermont.

The two hard-charging politicians, both 39, represent a rejection of centrist politics in the country’s largest swing state.

On Wednesday, Mr. DeSantis drew accusations of using a racist dog whistle after saying in a Fox News interview that voters should not “monkey this up” by electing Mr. Gillum, who, if he prevails, would be Florida’s first black governor. (Read more about that incident here.)

A spokesman for Mr. DeSantis denied that the candidate’s comments had a racial undertone, while a spokesman for Mr. Gillum referred to remarks by a state party official, who called the comments “disgusting.”