ASHBURN, Va. — Donald Trump’s feud with a Muslim Gold Star family may be driving GOP leaders to fits, but the Republican presidential nominee’s supporters aren’t fazed at all, saying they remain firmly behind the billionaire businessman and his tough talk on Twitter.

Mr. Trump drew a huge crowd to a rally Tuesday in the Virginia suburbs of Washington — and also attracted a sizable number of protesters dressed in traditional Muslim clothes, who said he’s gone too far in attacking their religion.

Steve Meyer, 52, echoed the views expressed by many Trump supporters at the rally when he defended Mr. Trump’s use of Twitter and blamed the news media for creating the feud with Khizr and Ghazala Khan, the Pakistani-born parents of a decorated Muslim U.S. soldier killed in Iraq who are speaking out against Mr. Trump.

Mr. Meyer, an engineer in Ashburn, also lashed out at Republican leaders who have sided with the Khans.

“To be honest, they are the old guard and part of the reason that Trump is seeing success,” he said. “We are tired of the Republican establishment sitting back and not doing anything except going along with the liberal agenda. We are also tired of the media criticizing our nominee while defending the Democrat nominee.”

Resistance from within the Republican Party and fierce opposition from minority groups have defined Mr. Trump’s primary run as a party outsider and populist candidate, and those conflicts are intensifying as he enters the general election race.

Trump supporter Kim Domin, who teaches special education students at an elementary school in Leesburg, said he shouldn’t tamper with success.

“I wish he was a tad less brash,” she said. “But he’s reached a lot of people by saying the things he’s saying. It’s gotten him here, hasn’t it?”

At the rally Mr. Trump didn’t mention the Khans or his proposed temporary ban on Muslim immigration and travel. Instead, he kept focused on top campaign issues of improving U.S. trade deals, keeping jobs in America and fighting Islamic State terrorists.

About 1,500 people filled the auditorium and about twice as many were turned away.

Mr. Trump took repeated shots at Democratic nominee Hillary Clinton. He hammered her for continuing to claim she didn’t send classified email on her private account as secretary of state, despite FBI Director James B. Comey recently testifying under oath that she did.

“She lied, pure and simple. She only knows to lie. It’s a big story. It’s become a big story,” he said, adding that the “dishonest” news media wasn’t holding her accountable.

In a speech at the Democratic National Convention, Mr. Khan denounced Mr. Trump for proposing a temporary ban on Muslims entering the U.S. to help prevent terrorism, questioning whether Mr. Trump had ever read the U.S. Constitution and accusing him of having made no sacrifices for the country.

Mr. Trump responded with a tweet that Mr. Khan had “no right” to attack him and with an interview saying he had sacrificed for the nation by building successful businesses.

The dispute quickly escalated as the Khans made the rounds on TV news shows and Mr. Trump continued to weigh in on Twitter.

“Captain Khan, killed 12 years ago, was a hero, but this is about RADICAL ISLAMIC TERROR and the weakness of our ‘leaders’ to eradicate it!” Mr. Trump tweeted Sunday.

Republican leaders who broke with Mr. Trump included House Speaker Paul D. Ryan and Senate Majority Leader Mitch McConnell, as well as Sen. John McCain, the party’s 2008 presidential nominee.

Mr. McConnell said that Army Capt. Humayun Khan, who was killed by a bomb in Iraq in 2004, was “an American hero” and that a Muslim travel ban was “simply contrary to American values.”

In a series of tweets, Mr. Trump said the Khans were diverting attention from the real issue of radical Islamic terrorism, and he criticized Mr. Khan for his “vicious attacks.”

He also questioned why Mrs. Khan remained silent on stage at the convention, apparently suggesting that the family’s Muslim religion kept her in a subordinate position to her husband.

The Muslim demonstrators took offense at most of Mr. Trump’s statements regarding the Khans.

“He said a lot of racist things to Muslims and hijabis,” said Kaamesa Hasan, 20, who wore a hijab head covering to the Trump rally at Briar Woods High School, where the line to get in stretched around the building.

Inside the school auditorium, a man wearing a keffiyeh scarf was escorted out by security shortly before Mr. Trump took the stage.

During Mr. Trump’s speech, several groups of young men supporting Islam staged demonstrations and were escorted out by security. The protests were peaceful, such as a man standing on a chair and raising his fist in the air.

Fawad Rahimayar, 19, was among a half-dozen protesters who stood up during the speech in front of a bank of TV cameras. He wore a T-shirt that read: “Islam means peace.” They were led out by security.

“His hateful rhetoric has obviously tainted the Muslim community,” Mr. Rahimayar said after the event.

He said many of the protesters were associated with the All Dulles Area Muslim Society (ADAMS) Center, a nonprofit community and organizing center for large Muslim population in Northern Virginia.

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