In a growing row between Donald Trump and the parents of a Muslim US Army officer killed in the Iraq war, the father said the Republican presidential nominee lacked the empathy needed to be a leader.
The dispute has dominated the election coverage since Khan gave a speech at the Democratic National Convention on Thursday night, with his wife standing at his side.
In the latest retorts, Trump responded 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 Khans' son, Army Captain Humayun Khan, was killed by a bomb in Iraq in 2004. His father spoke emotionally of the sacrifice his son had made for the country as an American Muslim, specifically criticising Trump's proposal to ban Muslims from entering the country.
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 conservative Muslims. "She had nothing to say, maybe she was not allowed to speak, you tell me," he said on Sunday.
But Khan retaliated on Monday: "There was no need to comment the way he commented," he said. "That initiated this conversation."
Ghazala Khan wrote an opinion piece in the Washington Post on Sunday saying that she had remained silent during her husband's remarks to cope with making her grief public during the convention.
The parents have appeared on television numerous times over the weekend and several leading Republicans have weighed in to express support for the family and to honour their sacrifice.
In interviews on Monday, the Khans spoke about their son and described the outpouring of support they have received during their very public exchange with Trump. Ghazala Khan said on CNN her family had always felt safe and protected as Muslims in the United States.
In response, Trump has tried to shift focus from the Khans.
"This story is not about Mr Khan, who is all over the place doing interviews, but rather RADICAL ISLAMIC TERRORISM and the US. Get smart!" Trump said on Twitter.
On Sunday, Democratic rival Hillary Clinton said Trump had scapegoated the parents. Leading Republicans, including House Speaker Paul Ryan and Senate Majority Leader Mitch McConnell, issued statements in support of the family.
Trump drew similar opprobrium a year ago when he said US Senator John McCain, who was taken prisoner for five years during the Vietnam War, was not a hero since he had been captured.
McCain, a leading voice in the party on military issues, on Monday condemned Trump for disparaging a fallen soldier's parents, saying in a statement his remarks do not represent the views of "our Republican Party."
In an open letter, nearly a dozen so-called Gold Star families - families who lost relatives in wars - said Trump cheapened their sacrifice and called for an apology.
"This goes beyond politics. It is about a sense of decency," it said. "That kind decency you mock as 'political correctness.'"
Additional reporting by Reuters.