Khan told CNN's Jim Acosta on "State of the Union" that he hopes Trump's family will "teach him some empathy."
"He is a black soul, and this is totally unfit for the leadership of this country," Khan said. "The love and affection that we have received affirms that our grief -- that our experience in this country has been correct and positive. The world is receiving us like we have never seen. They have seen the blackness of his character, of his soul."
Khan moved into the national spotlight after he pulled out a pocket copy of the Constitution during his speech at the Democratic National Convention. He said Trump would have barred his Muslim family from entering the United States.
Khan said Sunday Trump's "policy, his practices, do not reflect that he has any understanding of the basic, fundamental constitutional principles of this country."
"He talks about excluding people, disrespecting judges, the entire judicial system, immigrants, Muslim immigrants. These are divisive rhetoric that are totally against the basic constitutional principles," he said.
He also said Trump lacks key traits that presidents need.
"Two things are absolutely necessary in any leader or any person who aspires, wishes, to be a leader. That is moral compass and second is empathy," Khan said.
Khan called on House Speaker Paul Ryan and Senate Majority Leader Mitch McConnell to withdraw their support for Trump.
"It is a moral obligation -- history will not forgive them," he said. "This election will pass, but history will be written. The lack of moral courage with remain a burden on their souls."
He said those GOP leaders have a "moral, ethical obligation to not worry about the votes but repudiate him; withdraw the support. If they do not, I will continue to speak."
Both McConnell and Ryan issued statements Sunday afternoon praising Khan's son and reasserting their opposition to Trump's proposed temporary ban on Muslim immigration, but neither mentioned Trump.
"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," McConnell said. "All Americans should value the patriotic service of the patriots who volunteer to selflessly defend us in the armed services. 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."
Ryan said: "America's greatness is built on the principles of liberty and preserved by the men and women who wear the uniform to defend it. As I have said on numerous occasions, a religious test for entering our country is not reflective of these fundamental values. I reject it. 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."
The weekend back-and-forth drew a sharper response from Jeb Bush, one of Trump's vanquished GOP primary rivals.
"This is so incredibly disrespectful of a family that endured the ultimate sacrifice for our country," the former Florida governor tweeted Sunday night.
Former President George W. Bush's office, in a statement from a spokesman, echoed some of the Republican congressional leaders' views.
"President Bush isn't commenting on this presidential election. President Bush remains deeply grateful for the sacrifice of all Gold Star families, as we all should be. He thinks about them and prays for them each and every day."
Khan pushed back on Trump's suggestion that his wife, who was also on stage at the DNC, was not allowed to speak. He said she has high blood pressure and didn't want to speak for fear she wouldn't be able to hold herself together discussing her Gold Star son on stage.
"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 is why I showed him (the) Constitution. Had he read that, he would know the status a Gold Star mother holds in this nation."
Trump had first suggested Khan's wife was not allowed to speak in an interview with The New York Times' Maureen Dowd
, saying: "I'd like to hear his wife say something."
Then, he told Stephanopoulos, "If you look at his wife, she was standing there, she had nothing to say, she probably -- maybe she wasn't allowed to have anything to say, you tell me."
Khizr Khan's wife, Ghazala Khan, responded to Trump in a Washington Post op-ed Sunday
"Walking onto the convention stage, with a huge picture of my son behind me, I could hardly control myself. What mother could? Donald Trump has children whom he loves. Does he really need to wonder why I did not speak?" she wrote.
"Donald Trump said that maybe I wasn't allowed to say anything. That is not true," Ghazala Khan wrote. "My husband asked me if I wanted to speak, but I told him I could not. My religion teaches me that all human beings are equal in God's eyes. Husband and wife are part of each other; you should love and respect each other so you can take care of the family."
Khizr Khan had also said in an interview with The Washington Post
that Trump's attack on his wife was "typical of a person without a soul."
"Emotionally and physically -- she just couldn't even stand there, and when we left, as soon as we got off camera, she just broke down," Khan told the Post. "And the people inside, the staff, were holding her, consoling her. She was just totally emotionally spent. Only those parents that have lost their son or daughter could imagine the pain that such a memory causes."
Trump, in a statement released Saturday by his campaign, called Capt. Khan "a hero to our country and we should honor all who have made the ultimate sacrifice to keep our country safe."
But after Khan's Sunday CNN interview, Trump tweeted: "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!"
Earlier Sunday, he also tweeted: "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!"
Trump also pushed back on Khan's on-stage suggestion that he has "sacrificed nothing and no one" in an interview with with ABC News' George Stephanopoulos aired Sunday on "This Week."
"I think I've made a lot of sacrifices. I work very, very hard," Trump said. "I've created thousands and thousands of jobs, tens of thousands of jobs, built great structures. I've had tremendous success. I think I've done a lot."
And on Sunday night Trump's running mate, Indiana Gov. Mike Pence, issued a statement praising Humayun Khan and explaining Trump's policies.
"Donald Trump and I believe that Captain Humayun Khan is an American hero and his family, like all Gold Star families, should be cherished by every American," Pence said.
"Captain Khan gave his life to defend our country in the global war on terror. Due to the disastrous decisions of Barack Obama and Hillary Clinton, a once stable Middle East has now been overrun by ISIS. This must not stand," Pence added.
"By suspending immigration from countries that have been compromised by terrorism, rebuilding our military, defeating ISIS at its source and projecting strength on the global stage, we will reduce the likelihood that other American families will face the enduring heartbreak of the Khan family."
"Donald Trump will support our military and their families and we will defeat the enemies of our freedom."
Hillary Clinton also weighed in on Trump's Khan comments on Sunday in remarks to parishioners at the Imani Temple Ministries in Cleveland, Ohio.
"Mr. Khan 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 religions freedom, religious liberty," she said.
Clinton added: "I don't begrudge anyone of any other faith or of no faith at all. But I do tremble before those who would scapegoat other Americans, who would insult people because of their religion, their ethnicity, their disability, that is just not how I was raised, that is not how I was taught in my church."