"Two things are absolutely necessary in any leader or any person that aspires, wishes to be a leader. That is moral compass and, second, is empathy. This candidate is void of both traits that are necessary for the stewardship of this country," Khan said on CNN's "State of the Union."
"He is a black soul. And this is totally unfit for the leadership of this beautiful country," Khan said.
Speaking on NBC's "Meet the Press," Khan said the U.S. political process is "wonderful" and "beautiful" and urged patriotic people not to vote for Trump.
"I implore those patriotic Americans that would probably vote for Donald Trump in November, I appeal to them, not to vote for hatred, not to vote for fearmongering. Vote for unity. Vote for the goodness of this country. Vote for liberty and freedom. So my appeal now is to the folks that are going to be thinking to vote for Trump," Khan said.
Khan and his wife, Ghazala, appeared last week at the Democratic National Convention, where Khan gave an emotional speech in which he questioned whether Trump had ever read the U.S. Constitution and offered to lend him his copy. The Khans' son, Army Capt. Humayun Khan, was killed in Iraq in 2004. Khan said the Republican nominee had "sacrificed nothing and no one."
Trump lashed out at the Khans on Saturday, accusing Khizr Khan of being controlled by the campaign of Democratic nominee Hillary Clinton. Trump pointed to the sacrifices he has made as a businessman Saturday, saying he worked hard and employed thousands of people.
The backlash to Trump's remarks continued from both Democrats and Republicans on Sunday.
"Mr. Khan," Clinton said at a church in Cleveland, "paid the ultimate sacrifice in his family, didn't he. And what has he heard from Donald Trump? Nothing but insults, degrading comments about Muslims, a total misunderstanding of what made our country great, religious freedom, religious liberty. It's enshrined in our Constitution, as Mr. Khan knows, because he's actually read it."
Sen. Kelly Ayotte (R-N.H.), whose husband is a military veteran, called Humayun Khan a "true American hero" and slammed Trump for equating his sacrifices to those of the family of a soldier killed in action.
"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," Ayotte said.
"I am 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," she said, using the term for a military family that has lost a loved one.
In an interview with the New York Times, Sen. Lindsey O. Graham (R-S.C.), a veteran, said Trump's comments about the Khans were unacceptable.
"This is going to a place where we’ve never gone before, to push back against the families of the fallen,” he told the 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 said. “If you’re going to be leader of the free world, you have to be able to accept criticism, and Mr. Trump can’t.”
Khizr Khan's wife, Ghazala, did not speak onstage. The Khans are Muslim, and Trump questioned why Ghazala Khan did not speak. "She probably — maybe she wasn’t allowed to have anything to say,” he said.
In a subsequent interview, Ghazala Khan said she did not speak because she becomes overwhelmed with emotion whenever she sees her son's picture. (A large photo of Humayun Khan was displayed on the video screen onstage during her husband's speech.)
In an opinion piece published Sunday on The Washington Post's website, Ghazala Khan responded directly to Trump.
"Donald Trump has asked why I did not speak at the Democratic convention. He said he would like to hear from me. Here is my answer to Donald Trump: Because without saying a thing, all the world, all America, felt my pain. I am a Gold Star mother. Whoever saw me felt me in their heart," she wrote.
She said Trump is "ignorant" when he talks about Islam. Trump has called for banning Muslims from entering the United States.
"If he studied the real Islam and Koran, all the ideas he gets from terrorists would change, because terrorism is a different religion." she wrote. "Donald Trump said he has made a lot of sacrifices. He doesn’t know what the word sacrifice means."
On Sunday, Khizr Khan defended his wife's silence at the convention, telling CNN that she is ill, has high blood pressure and cannot keep herself composed when she sees images of her son.
"For this candidate for presidency to not be aware of the respect of a Gold Star mother standing there, and he had to take that shot at her, this is height of ignorance," Khan said.
"This country holds such a person in the highest regard. And he has no knowledge, no awareness. That is the height of his ignorance. She is ill. She had high blood pressure. People that know her looked at her face, and she said, "I may fall off the stage,'" said Khan, his voice quavering with emotion. "And I told her that you have to assemble yourself and stand for the beauty of this tribute that is being paid."
In a tweet Sunday, Trump responded to Khan.
"I was viciously attacked by Mr. Khan at the Democratic Convention. Am I not allowed to respond? Hillary voted for the Iraq war, not me!" Trump wrote.
Khan said McConnell and Ryan must speak out against Trump.
It is their "moral, ethical obligation to not worry about the votes but repudiate him, withdraw the support. If they do not, I will continue to speak, and I am speaking," Khan said.
In statements Sunday, Ryan and McConnell expressed support for the Khans and reiterated their opposition to Trump's proposed ban on Muslims, but did not abandon their support of the Republican nominee.
“Captain Khan was an American hero, and like all Americans I’m grateful for the sacrifices that selfless young men like Capt. Khan and their families have made in the war on terror. All Americans should value the patriotic service of the patriots who volunteer to selflessly defend us in the armed services," McConnell said in a statement.
"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,” McConnell said.
Ryan said a religious test for people coming into the country is "not reflective" of the nation's values.
"Many Muslim Americans have served valiantly in our military, and made the ultimate sacrifice. Captain Khan was one such brave example. His sacrifice — and that of Khizr and Ghazala Khan — should always be honored. Period," Ryan said in a statement.
In a statement released Saturday night, Trump called Humayun Khan a "hero" and accused Khizr Khan of saying "inaccurate things" about him at the convention.
In a tweet on Sunday morning, Trump said: "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!"
In his CNN interview, Khizr Khan forcefully denounced Islamist terrorism.
"What he cites in the name of Islam and all that, that is not Islam at all. I wish he would have — somebody would have put something in his head that these are terrorists, these are criminals. These folks have nothing to do with Islam," Khan said.
"Muslims hate this menace of terrorism as much as any other place. It is our duty to keep this country, our country, beautiful country, safe," he said.
Anne Gearan contributed reporting from Cleveland.