WASHINGTON - Donald Trump's confrontation with the parents of a slain Muslim American soldier escalated Monday, shaking the US presidential campaign amid Republican Party outcries over the real estate tycoon's comments about the couple.

The feud has become a flash point of the 2016 White House race that pits Trump against Democrat Hillary Clinton.

It has dominated the news cycle ever since Pakistani immigrant Khizr Khan galvanized the Democratic National Convention last Thursday with a tribute to his dead son in which he rebuked the Republican nominee for having "sacrificed nothing" for the country.

The father of slain soldier said on Monday that the Republican presidential nominee lacked the empathy to be a leader and chided him for throwing the first salvo in their exchange.

Trump responded simultaneously on Twitter to Khizr and Ghazala Khan's morning television appearances, saying he was being "viciously" attacked.

Asked on CNN what message he could give Trump, Khizr Khan said he wanted to maintain his family's dignity and convey to Trump "that a good leader has one trait ... empathy."

"It is basic character, realizing, feeling the pains, the difficulties of the people that you wish to lead," Khan said. "And that is missing."

The issue has dominated the election campaign in recent days after Khan spoke at the Democratic National Convention on Thursday night, with his wife standing at his side.

In response to the speech, Trump said Ghazala Khan might not have been "allowed" to speak, implying her silence reflected restrictions placed on women by some traditional Muslims.

"There was no need to comment the way he commented," Khan said on Monday. "That initiated this conversation."

The Council on American-Islamic Relations (CAIR), a leading Muslim civil rights and advocacy organisation, has demanded that Trump’s apologise.

In a statement responding to Trump's remarks on Sunday night, CAIR Board Chairperson Roula Allouch said: "As the leader of America's largest Muslim civil rights organisation, I urge Donald Trump to apologise for his shameful remarks disparaging a Muslim Gold Star family and for his repeated use and promotion of anti-Muslim stereotypes.

"Just as Donald Trump must apologise for his un-American remarks, Republican Party leaders must finally repudiate their candidate's divisive rhetoric."

The families of 23 other slain US soldiers have also called on Trump to apologise for his "repugnant and personally offensive" remarks.

"We feel we must speak out and demand you apologise to the Khans, to all Gold Star families, and to all Americans for your offensive, and frankly anti-American, comments," they said in an open letter.

Army captain Humayun Khan was killed by a suicide bomber in Iraq in 2004.

Trump's sustained hostility toward the Khans - alarming in part because criticism of Gold Star families of war dead has traditionally been off limits in American political discourse - has received bipartisan condemnation including from Senator John McCain, a former prisoner of war once mocked by Trump for being captured in Vietnam.

"I cannot emphasise enough how deeply I disagree with Mr. Trump's statement," McCain said in a lengthy statement.

"While our party has bestowed upon him the nomination, it is not accompanied by unfettered license to defame those who are the best among us," he added. "I hope Americans understand that the remarks do not represent the views of our Republican Party, its officers, or candidates."

Unequivocal condemnation also rained down from Republican House Armed Services Committee chairman Mac Thornberry, who said he is "dismayed at the attacks" on the Khans.

The top two Republicans in Congress released carefully crafted statements denouncing the remarks, although without mentioning the name of Trump, who has urged a ban on Muslims entering the country.

Senate Majority Leader Mitch McConnell said Sunday in the thick of the feud that "Captain Khan was an American hero."

"And as I have long made clear, I agree with the Khans and families across the country that a travel ban on all members of a religion is simply contrary to American values."

House Speaker Paul Ryan also said he rejects such a religious test. Speaking of Captain Khan, Ryan said "his sacrifice - and that of Khizr and Ghazala Khan - should always be honored. Period."

SPECIAL CORRESPONDENT/Agencies