President Barack Obama also joined the fray later on Monday, implicitly rebuking Trump for his comments.
Khizr Khan, the soldier's father, was a guest on "New Day" on CNN Monday with his wife Ghazala Khan, and delivered the criticism of Trump amid an extended plea for unity and empathy. He stressed the need to work with Muslim communities to combat radicalization, while again slamming Trump for his divisive rhetoric.
"Communities coming together is the solution. We are as concerned as Donald Trump is about the safety of this country. We are a testament to the goodness of this country," he said. "We need a leader that will unite us, not disrespect, not by derogatory remarks. I feel bad about the discourse that this campaign, this election campaign has taken."
"That's all I wish to convey to him. That a good leader has one trait -- earlier I said -- empathy."
Though Obama's comments to veterans in Atlanta Monday afternoon didn't mention Trump by name, the President seemed to offer support to Khan's perspective.
"No one has given more for our freedom and our security than our Gold Star families," Obama said, adding that those families "represent the very best of our country."
"They continue to inspire us every day and every moment," Obama said. "They serve as a powerful reminder of the true strength of America."
Khan, in his appearance on "New Day," exhorted Trump to take policy steps that would curtail extremism, not exacerbate it.
"Let's make those policies and implement them. But let's not create, strengthen the hands of enemies by these remarks, by these statements. Believe me, these statements strengthen their recruitment process," he said.
Khan added that he's received an outpouring of support in response to the controversy, highlighting his "email box full of messages from prominent Republicans" who have criticized Trump for his remarks.
"The amount of recognition, the amount of love, support, care -- street corners, any place we go to -- people come up and say, you have spoken what we wanted to say," Khan said. "I am amazed at the love and care that we have received."
The family also insisted that they did not seek out the confrontation with Trump, describing themselves as "private people" and saying "we want to be out of this controversy."
"We don't want to continue. That is not our style. We are a decent, dignified family of this country, very appreciative of the blessings that we have enjoyed...this is not our path," Khan said of the high-profile conflict.
But even as the Khans expressed their desire to leave the controversy behind, Trump responded to the criticism during the interview on Twitter. He wrote, "Mr. Khan, who does not know me, viciously attacked me from the stage of the DNC and is now all over T.V. doing the same - Nice!"
Trump also tweeted that the focus should be on "radical Islamic terrorism," not the Khans.
Asked on "New Day" to respond to Trump's tweet, Khan said, "I spoke what was appropriate, and if he is watching, just imagine, there was no need to comment the way he commented. That initiated this conversation. I again say, we want to maintain our dignity. We want to maintain my family's dignity, my son's dignity and sacrifice. And he should listen to America, what America and the world is telling about the remarks, about the lack of empathy, and that's all I wish to convey to him."
Khan also said on "New Day" that Republicans needed to go further and disavow Trump completely, calling him "ignorant" and "arrogant" and noting that "he has been abusing, disrespecting women, judges, all decent Americans."
"Enough is enough. Every decent Republican ... has rebuked this behavior, yet no one has stood up and said, enough, stop it. You will not be our candidate."
Joel Benenson, a senior adviser to the Clinton campaign, said on "New Day" Monday that Trump was playing the victim.
"To claim Donald Trump is being attacked, playing the victim, this a man that is not only tone-deaf, but temperamentally and completely unfit to be commander in chief," Benenson said.
Khan also explained that despite the family's desire for privacy, they decided to participate in the Democratic National Convention "because a tribute was being paid and there was context to my conversation. We had been patiently subjected to the maligning of this candidate for a whole year."
Later in the interview, Ghazala Khan again responded to Trump's criticism of her silence at the convention, pointing to her op-ed in the Washington Post about the issue and adding, "I can say that my religion or my family or my culture never stopped me saying whatever I want to say."
Discussing her appearance at the DNC, Ghazala Khan explained, "I have done very well saying my mind out, but that time was different. And anybody can see it was different that time when I was standing there in front of America. Without saying a word, I had lots of love. I touched lots of hearts. So I'm thankful for everything that America has given me."
Trump's criticism of the Khans sparked a major attack from the Clinton campaign -- and a severe backlash within his own party.
On Monday morning, Sen. John McCain issued a sharply-worded statement condemning Trump's remarks and calling on the GOP nominee to "set an example for our country and the future of the Republican Party."
In the statement, McCain wrote: "In recent days, Donald Trump disparaged a fallen soldier's parents. He has suggested that the likes of their son should not be allowed in the United States -- to say nothing of entering its service. I cannot emphasize enough how deeply I disagree with Mr. Trump's statement. I hope Americans understand that the remarks do not represent the views of our Republican Party, its officers, or candidates."
He added, "In the end, I am morally bound to speak only to the things that command my allegiance, and to which I have dedicated my life's work: the Republican Party, and more importantly, the United States of America. I will not refrain from doing my utmost by those lights simply because it may benefit others with whom I disagree."
In an implicit rebuke of Trump, the White House said Monday that Gold Star families deserve only "honor and gratitude" for their loved ones' service.
Speaking aboard Air Force One, spokesman Eric Schultz wouldn't provide a specific response to Trump's comments about the Khan family. But he said honoring Gold Star families should rise "above politics."
"This is something the president believes should rise above politics," he said. "The sacred covenant we have with the brave men and women who serve our country, and those who perform the ultimate sacrifice, should be one that we honor every single day. The president believes that no one has given more for our country's security, for our country's safety, for our country's freedom than Gold Star families."
"Families who make the ultimate sacrifice for this country's freedom and this country's safety deserve nothing but our country's honor and gratitude and deepest respect," Schultz said.
Another Republican Senator -- Roy Blunt of Missouri -- also issued a statement Monday admonishing Trump and praising the Khans.
Blunt wrote, "I remember how much I worried about my son Matt during his years of active duty. The Khans have made the greatest possible sacrifice for our country; they deserve to be heard and respected. 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."
Texas Rep. Mac Thornberry, chairman of the House Armed Services Committee, also issued a statement saying he was appalled by attacks on the Khans. Without mentioning Trump by name, the Republican lawmaker said, "I am dismayed at the attacks Khizr and Ghazala Khan have endured after they spoke about their son's service and sacrifice ... I believe that each of us are called every day to show our deepest respect and gratitude to all of those who protect our freedom and their families."
Democratic presidential candidate Hillary Clinton on Sunday said she didn't know how low Trump could go.
"To have Trump do what he did, I don't know where the bounds are," Clinton said. "I don't know where the bottom is."
Republican also tore into the Republican presidential hopeful, with Sen. Lindsey Graham, R-South Carolina, saying in a statement, "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."
"This is so incredibly disrespectful of a family that endured the ultimate sacrifice for our country," Jeb Bush, a Trump rival in the 2016 GOP primary, said on Twitter Sunday evening.
The Khans, whose son Army Capt. Humayun Khan was killed in Baghdad in 2004, took the stage Thursday night at the Democratic National Convention, where Khizr Khan rejected Trump's proposal to temporarily ban Muslims from entering the United States as unconstitutional, pulling a copy of the Constitution from his breast pocket, questioning if Trump has ever read it, and saying that Trump has "sacrificed nothing and no one."
Trump responded by criticizing wife Ghazala Khan's silence and suggesting she wasn't allowed to speak.
Republican congressional leaders also took issue with Trump, issuing statements Sunday that praised the Khan family and reaffirmed their opposition Trump's proposed Muslim travel ban.
House Speaker Ryan, R-Wisconsin, said in a statement, "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."
Khizr Khan told CNN's Jim Acosta over the weekend that 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."
On Sunday, Trump responded to the uproar.
"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 wrote in the first of two tweets.
"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 said in his second tweet.
As Trump pushed back, first asking whether Ghazala Khan's silence on stage was related to her faith, Khizr Khan again attacked Trump Sunday on CNN's "State of the Union."
"He is a black soul, and this is totally unfit for the leadership of this country," Khan said.