He called questions about the tweet “a vile attempt to demean” Mr. Alembik and the DeSantis campaign and asserted that if someone of color had said the same about Mr. Obama, it wouldn’t be considered racist.

Two recent polls show Mr. DeSantis, 40, who resigned from Congress this month to focus on his campaign, on the heels of the Democratic candidate, Mayor Andrew Gillum of Tallahassee, who would be Florida’s first black governor.

The day after he won the Republican primary on Aug. 28, Mr. DeSantis was accused of using a racist dog whistle during an interview on Fox News. Mr. DeSantis painted Mr. Gillum as a far-left socialist who wants to raise taxes and bankrupt the state.

“The last thing we need to do is monkey this up by trying to embrace a socialist agenda,” he said.

Mr. Gillum, 39, does not identify as a socialist, and he and other Florida Democrats strongly criticized Mr. DeSantis’s choice of words.

“In the handbook of Donald Trump, they no longer do whistle calls,” Mr. Gillum said at the time. “They’re now using full bull horns.”

Later that week, racist robocalls targeting Mr. Gillum were placed to residents from an Idaho-based white supremacist group.

In the audio of one robocall obtained by The New York Times, a man pretending to be Mr. Gillum can be heard talking in the exaggerated accent of a minstrel performer. “Well hello there,” it begins, “I is Andrew Gillum.” He then talks about mud huts and unfair policing practices, and asks for the listener’s vote amid sounds of drums and monkeys.