Inside a convenience store in Littleton, N.H., Gov. Ron DeSantis of Florida was confronted on Thursday with both sides of the Israeli-Palestinian conflict.

It was just one interaction on one day in a long campaign. But it could well be a preview of how the divisive issue could play out in the presidential election.

Mr. DeSantis, standing next to a row of coffee dispensers inside Simon’s Market, began his campaign stop by telling a group of voters that he had just issued an executive order to help bring stranded Americans home from Israel.

Laurie Anan, an undecided voter who said she had visited Israel in August, broke in to thank Mr. DeSantis, saying that the images of the bloody attacks by Hamas over the weekend were “devastating.”

“We’re happy to do it,” Mr. DeSantis said, before going on to describe the brutality of the execution of Israeli civilians and criticize the Palestinian residents of Gaza.

“The people there had an opportunity to make something,” Mr. DeSantis said. “And they decided to cast their lot in with Hamas. And so it’s created this dysfunctional society.”

That led Ron Lahout — an Arab American who said he had voted for both Barack Obama and Donald J. Trump — to interrupt, kicking off a lengthy back-and-forth between the two men, one the governor of the nation’s third-largest state, the other the owner of a local ski shop.

“Well, Ron, what do you think about the annihilation and the decapitation of all the Palestinians in Gaza right now?” asked Mr. Lahout, 65, a lifelong New Hampshire resident who said he had worked in refugee camps in Gaza in the 1980s.

“They are not decapitating babies’ heads,” Mr. DeSantis replied of the Israeli armed forces, referring to unverified reports about Hamas atrocities. “They are not intentionally doing that.”

“They are blowing up entire residential buildings,” said Mr. Lahout, who later described himself in an interview as a Republican eager to vote for anyone but President Biden.

The tense but respectful exchange lasted for nearly two minutes, with Mr. Lahout calling Gaza a “prison” and Mr. DeSantis saying that Hamas was using Palestinian civilians as “human shields” and crediting Israel for trying to warn local populations ahead of its strikes.

Mr. DeSantis has generally expressed little sympathy for Palestinian civilians in Gaza, while urging Israel to use “overwhelming force” and wipe Hamas “off the face of the earth.” He has said there is no “moral equivalence” between the attacks by Hamas and Israel’s response. And he has criticized Mr. Trump, the front-runner for the Republican nomination, for criticizing Israel’s prime minister, Benjamin Netanyahu, and has ordered Florida to send charter flights to Israel to bring Americans home.

At the convenience store, Mr. Lahout insisted that he did not “condone the killing of any innocent civilians,” in what resembled a closing statement.

“And I don’t condone what Hamas did in the kibbutzes,” he continued. “But Israel is doing the exact same thing with Benjamin Netanyahu, who is a radical, right-wing crazy person. And I see hundreds of Palestinian families that are dead, and they have nowhere to go because they can’t leave Gaza because no one’s opening their borders.”

“Well, but that’s the thing,” Mr. DeSantis countered. “You bring up a good point though. You bring up a really good point. Why aren’t these Arab countries willing to absorb some of the Palestinian Arabs? They will not do it — Egypt will not do it, Saudi Arabia will not do it. None of them will do it.”

Mr. Lahout delivered a final retort: “You had my vote, but you don’t now.”

With that, he turned on his heels and stormed out of the store, and Mr. DeSantis resumed fielding questions from voters.