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But if the debate was the marquee event of a campaign for the Texas governor’s mansion that is likely to cost more than $100 million, it did not seem to deliver a key moment that would significantly propel or hobble either candidate. That outcome appeared likely to benefit Mr. Abbott, who has been leading in the polls and has commanded a larger campaign war chest going into the final stretch.

The hectic pace of the exchanges — strictly limited by the moderators to 30 or 60 seconds — devolved at times into rhetorical finger-pointing between the two politicians over whose beliefs, diametrically opposed, were more outside the mainstream.

“Beto’s position is the most extreme,” Mr. Abbott said, suggesting that his Democratic rival supported allowing abortions at any point in a pregnancy.

“It’s completely a lie,” Mr. O’Rourke responded. “I never said that, and no one thinks that in the state of Texas.” He added: “He’s saying this because he signed the most extreme abortion ban in America. No exception for rape, no exception for incest.”

For weeks, the two candidates have clashed repeatedly on the airwaves, but they had yet to spar in person. Mr. O’Rourke tried to confront Mr. Abbott during a news conference in Uvalde after the massacre at an elementary school there in May, accusing him of doing nothing to prevent such violence, before Mr. O’Rourke was escorted out. Mr. Abbott did not respond at the time.

On Friday, Mr. Abbott similarly tried to ignore Mr. O’Rourke’s attacks as much as possible, rarely looking at his opponent as he spoke or listened.

Mr. O’Rourke went after Mr. Abbott from the start, blaming the governor’s “hateful rhetoric” for the killing of an undocumented migrant in West Texas this week and saying that the governor had “lost the right to serve this state” because of the police failures in the response to the Uvalde shooting.

Mr. Abbott repeatedly accused his challenger of misrepresenting facts. “He just makes this stuff up,” he said.

Mr. O’Rourke, a polished debater, appeared more at ease with the fast format of the debate. Mr. Abbott at times seemed to rush to make his points, and struggled with a question about whether he believed that emergency contraception was the “alternative” for someone who became pregnant from rape or incest in Texas, given that abortion is banned even in those cases.

“An alternative, obviously, is to do what we can to assist and aid the victim,” Mr. Abbott said. “They’re going to know that the state, through our Alternatives to Abortion program, provides living assistance, baby supplies, all kinds of things that can help them.”

It appeared clear that Mr. O’Rourke was the strongest challenger Mr. Abbott has had in his political career, stretching back into the 1990s. Mr. Abbott has never faced a primary opponent of note, and in his previous runs for governor, he easily swept aside Democratic opposition.

The hourlong debate was held in the border city of Edinburg, far from the large population centers of this increasingly urbanizing state but deep in the heart of Hispanic South Texas, where Mr. Abbott and Republicans have increasingly made inroads.

The location also put a spotlight on a topic that has been among the most effective issues for Mr. Abbott: the record numbers of unauthorized migrants continuing to arrive at the southern border.

The candidates, both in red ties, debated from a sitting position; Mr. Abbott has used a wheelchair since he was 26, when an oak tree fell on him while he was jogging, paralyzing him below the waist.

The candidates received no time for introductory comments and gave 30-second closing remarks, a format that played to Mr. Abbott’s strengths as a direct, often terse speaker, and limited Mr. O’Rourke’s tendency to build long rhetorical flourishes.

And the timing, on a Friday evening when many Texans are more consumed with high school football, appeared likely to reduce the number of people watching live.

Chris Evans, a spokesman for Mr. O’Rourke, said before the debate that the Abbott campaign had proposed the terms and would not accept any changes. “They declined to have voters in the audience,” he said. An Abbott spokesman, Mark Miner, said that Mr. O’Rourke was in “no position to run the state if he can’t even comprehend simple debate rules.”

Democrats in Texas have pinned their hopes on Mr. O’Rourke before, but so far he has managed only to be victorious in defeat. In his name-making 2018 run for Senate, he came within three percentage points of unseating Senator Ted Cruz, a strong showing in Republican-dominated Texas, but still a losing one.

Friday’s debate, just a few weeks before early voting begins in Texas, came at a crucial moment for both campaigns, especially Mr. O’Rourke’s. Over the summer, some polls had suggested a tightening race after the Uvalde killings and the Supreme Court’s decision overturning Roe v. Wade. But more recent surveys show Mr. Abbott more firmly in control, with a lead of about seven percentage points.

For Mr. O’Rourke, the former congressman from El Paso and a Democratic presidential candidate in 2020, the debate was a chance to recapture momentum and his most direct opportunity to prosecute his case against Mr. Abbott, a two-term incumbent who has led the state for eight years under unified Republican control of state government.

For Mr. Abbott, it was a night to make it through unscathed. His campaign had prepared for weeks for the encounter, seeing Mr. O’Rourke as a skilled debater with significant experience from his run for president in 2020.

The governor navigated tough questions, including one that in many ways launched Mr. O’Rourke into this race: the failure of the energy grid last February. Mr. Abbott stuck to his talking points — “the grid is more resilient and reliable than it’s ever been,” he said — and Mr. O’Rourke did not appear able to capitalize on the exchange.

“It seemed pretty even, and a bit of a tie probably benefits Governor Abbott in this case,” said Álvaro J. Corral, an assistant professor of political science at the University of Texas Rio Grande Valley in Edinburg. “I didn’t see a moment that indicated a change in that underlying dynamic.”

As the campaign entered its final months, Mr. Abbott has pressed his fund-raising advantage — $45 million on hand as of the last filing in mid-July, versus $23 million for Mr. O’Rourke — and went statewide with television ads at least two weeks before Mr. O’Rourke did so. Now both campaigns are bombarding Texans with messages, often negative, on television and online.

Before the debate, Mr. O’Rourke held a news conference in a playground in Edinburg with several parents and relatives from Uvalde whose children were shot and killed at Robb Elementary School. The families rode a bus together from Uvalde that morning to press for action on gun control; Mr. O’Rourke criticized Mr. Abbott for setting rules that would not allow the parents to watch from inside the hall.

The massacre took up significant time early in the debate. Mr. O’Rourke, who during the 2019 campaign urged taking away AR-15-style rifles after a deadly mass shooting in El Paso, emphasized his moderated position, calling to raise the age to buy an AR-15 to 21 from 18.

Mr. Abbott said that was unrealistic, citing recent court decisions striking down gun restrictions.

“We want to end school shootings,” he said. “But we cannot do that by making false promises.”

Reid J. Epstein contributed reporting.