Prominent Republicans have condemned Donald Trump over his comments about Khizr and Ghazala Khan, the parents of an American soldier who was killed in Iraq.

John McCain, the GOP presidential nominee in 2008, challenged Trump on Monday “to set an example” and said: “I cannot emphasize how deeply I disagree with Mr Trump’s statement [about the Khans]. I hope Americans understand that the remarks do not represent the views of our Republican party, its officers, or candidates.”

Barack Obama, who beat McCain in the 2008 election, addressed the annual conference of Disabled American Veterans in Atlanta on Monday. He did not mention Trump by name, but he expressed sharp disapproval of any criticism of those who serve in the military and their families.

“As commander-in-chief, I’m pretty tired of some folks trash-talking America’s military and troops,” the president said. “No one – no one – has given more for our freedom and our security than our Gold Star families [those whose family members died in service].

“[They] have made a sacrifice that most of us cannot even begin to imagine. They represent the very best of our country. They continue to inspire us every day, every moment. They serve as a powerful reminder of the true strength of America. We have to do everything we can for those families and honor them and be humbled by them.”

In rebuking Trump, McCain joined a growing chorus of senior Republicans who have felt compelled to distance themselves once more from the nominee. Like other leading party figures, however, he did not rescind his endorsement of Trump for the presidency. The Arizona senator faces a tough re-election battle of his own in November.

Captain Humayun Khan died while protecting his army unit from a suicide bombing in 2004. Last Thursday, at the Democratic national convention in Philadelphia, his father, Khizr Khan, spoke about his opposition to Trump’s proposed ban on Muslim immigration.

This weekend, in an interview with ABC and on Twitter, Trump questioned the authorship of the speech, insisted he had made sacrifices for his country comparable to those of the Khans, and complained of being “viciously attacked” by Khizr Khan.

The Republican nominee also questioned why Ghazala Khan, the fallen soldier’s mother, stood silently onstage at the DNC – insinuating that she was not permitted to speak because of her faith.

Khizr Khan responded in a CNN interview on Sunday, saying Trump had a “black soul”. On Monday, Ghazala Khan told NBC, as she had written in the Washington Post a day before, that the reason she had remained silent on stage was that she was too overcome by grief over her son’s death to speak about it.

“I didn’t talk because I don’t have the heart when it comes to Humayun. I can’t,” she said. “I can’t talk – my words will be not that strong or I will start crying.”

On Monday, McCain, a decorated Vietnam veteran, thanked the Khans for coming to the US – they are of Pakistani heritage and their son was born in the UAE – and said their son’s death “was a shining example of the valor and bravery inculcated into our military”. While he did not revoke his support for Trump, who last summer mocked him for having been a prisoner of war in Vietnam, his criticism came with a sharp warning.

The New Hampshire senator Kelly Ayotte, one of the most endangered Republicans, was “appalled that Donald Trump would disparage them and that he had the gall to compare his own sacrifices to those of a Gold Star family. The Khan family deserves nothing less than our deepest support, respect and gratitude, and they have every right to express themselves in any way they choose.”

Marco Rubio told reporters on Monday he found Trump’s comments to be “unfortunate”.

“[Captain] Khan is a hero, for every measure,” the Florida senator, who decided to seek re-election to the legislature after the failure of his presidential campaign, said following an event in Tampa.

“I think his parents are heroes and they have a first amendment right to speak out on their politics, as all Americans do. Their son died for the Constitution and I honor that.”

Rubio pointed to his praise during the presidential debates of American Muslims who served in the military. He nonetheless added that it was “a complex issue”, because Khan had “lost his life in the hands of radical jihadists”.

“They took their view of their faith and used it to justify killing innocents and attacking them,” he said of Khan’s killers.

Rubio, who has embraced Trump and urged voters to elect him, also echoed his former opponent’s call to crack down on immigration from certain regions. While Rubio has said he does not support a ban on all Muslims, the senator said “it’s very difficult to vet people from certain parts of the world where there are not vetted governments”.

He added: “We live in a world now where we have to know exactly who’s coming in, why they’re coming, who they are, and what they’re up to.”

Some on Capitol Hill declined to comment specifically, instead urging Trump to refrain from veering off-message. Senator Roy Blunt, who is defending his seat in Missouri, said: “My advice to Donald Trump has been and will continue to be to focus on jobs and national security and stop responding to every criticism whether it’s from a grieving family or Hillary Clinton.”

Other Republicans in Congress who spoke out included Adam Kinzinger, a representative from Illinois and Iraq veteran who said it was “wholly unpatriotic” to attack service members and their families.

“No one can truly understand the sacrifice a family makes when their loved one serves their country or the heartbreak of losing a loved one,” Kinzinger, who reluctantly backed Trump, said in a statement. “There’s no greater love than a man who lays down his life for his friends.”

In his appearance on NBC’s Today Show, Khizr Khan said he was grateful that the Senate majority leader, Mitch McConnell, and House speaker, Paul Ryan, had condemned Trump’s comments over the weekend, but suggested they were not listening to the Republican party.

“I have an email inbox full of assurance that I am right, that we are right, and he needs to mend his way to deserve their vote,” Khan said.

That inbox included an email from a prominent Republican, Khan said, who said he would not be voting in the presidential election as Trump “has not qualified in their judgment to be deserving of their support”.

At the Democratic convention, Khan waved a pocket-sized US constitution and questioned whether Trump had ever read it. On Monday, Khan was asked about Trump’s claim that the mourning father had “viciously attacked” him by doing so. He laughed, saying the comments proved Trump was unfamiliar with the first-amendment protection of freedom of speech.

“This candidate amazes me,” Khan said. “His ignorance, he can get up and malign the entire nation, the religions, the communities, the minorities, the judges – and yet a private citizen in this political process, in his candidacy for the stewardship of this country, I cannot say what I feel?

“That proves the point.”

Lindsey Graham, the former Republican presidential candidate and South Carolina senator, said Trump’s words were unacceptable. “This is going to a place where we’ve never gone before, to push back against the families of the fallen,” Graham, who has not endorsed Trump, told the New York Times.

“There used to be some things that were sacred in American politics, that you don’t do, like criticizing the parents of a fallen soldier, even if they criticize you,” Graham added. “If you’re going to be leader of the free world, you have to be able to accept criticism, and Mr Trump can’t.”

Trump, who as a young man obtained deferments and did not serve in Vietnam, also faced criticism from the families of 17 Americans who died in war, who in an open letter asked him to apologize to the Khans and other families of fallen soldiers for comments they said were “repugnant, and personally offensive”. The Gold Star families said Trump’s statements minimized the risks people take by going to war.

“When you question a mother’s pain, by implying that her religion, not her grief, kept her from addressing an arena of people, you are attacking us,” the families wrote in a letter to Trump. “When you say your job building buildings is akin to our sacrifice, you are attacking our sacrifice.

“You are not just attacking us, you are cheapening the sacrifice made by those we lost.”