ATLANTA — It has become the “deplorables” moment of the Georgia governor’s race.

In a private conversation secretly recorded in May and made public on Monday, Casey Cagle, the Republican lieutenant governor, was captured criticizing the over-the-top tone of his party’s primary. He said that it had become focused on “who had the biggest gun, who had the biggest truck, and who could be the craziest.”

Now Mr. Cagle’s opponent in the July 24 Republican runoff for governor, Brian Kemp, is accusing Mr. Cagle of turning up his nose at the salt-of-the-earth Republican base — much as President Trump did when, as a candidate, he criticized Hillary Clinton for using the phrase “basket of deplorables” to describe many of his supporters.

Mr. Cagle’s campaign has insisted that he was referring to the candidates, not the voters. But the recording of Mr. Cagle underscored the dilemma for a Republican Party in which some leaders are privately put off by the rough and raucous style perfected by Mr. Trump — but fear that any criticism of that style might be interpreted as contempt for voters who are energized by it.

The recording of Mr. Cagle’s comments released by his opponent has added additional drama to what was already a neck-and-neck race between two seasoned, Trump-supporting politicians to succeed the current governor, Nathan Deal, a Republican. Mr. Cagle and Mr. Kemp are vying to prove who is further to the right during a primary season in which Republican voters in other nearby states, when offered a choice, have gone with the more conservative option.