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8 year oldMr McCain said Mr Trump did not have an "unfettered license to defame those who are the best among us".
Mr Trump has come under fire for criticising the Muslim American parents of a US soldier killed in Iraq.
Democratic lawmakers and the soldier's father have called on Republicans to disavow Mr Trump.
"I cannot emphasize enough how deeply I disagree with Mr. Trump's statement," Mr McCain said in a statement.
"I hope Americans understand that the remarks do not represent the views of our Republican Party, its officers, or candidates."
Soldier's mother: 'I was in pain'
Dead Muslim soldier's parents row with Donald Trump
Mother hits back at Trump over speech
Mr McCain, a war veteran and the party's 2008 presidential nominee, thanked the Khan family for immigrating to America, adding "we're a better country because of you."
Last year Mr Trump sparked a backlash after he said Mr McCain was not a war hero because he was captured in Vietnam.
The latest escalation comes as Republican leaders have stepped forward to rebuke Mr Trump's fight against the Gold Star parents.
In the US, parents who have lost a child in war are known as Gold Star families.
Reaction to Mr Trump's comments
The couple's son, US Army Capt Humayun Khan, was killed by a car bomb in 2004 in Iraq at the age of 27.
Pakistani-born Khizr Khan and his wife, Ghazala Khan, appeared at the Democratic National Convention in Philadelphia last week to speak about their son's sacrifice.
In an emotional speech, Mr Khan said his son would not even have been in America if it had been up to Mr Trump, who has called for a ban on Muslims entering the US.
In an interview with ABC's This Week, Mr Trump suggested Mrs Khan may have not been allowed to speak.
"She had nothing to say... Maybe she wasn't allowed to have anything to say. You tell me."
Mrs Khan fired back in opinion article for the Washington Post, saying she was too upset to speak at the convention.
She said Mr Trump was ignorant about Islam and did not know the meaning of the word sacrifice.
"Walking on to 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.
On Monday, family members of eleven service-members killed in action wrote to Mr Trump, expressing dismay at his treatment of the family of Humayun Khan, and demanding an apology.
"When you question a mother's pain, by implying that her religion, not her grief, kept her from addressing an arena of people, you are attacking us. When you say your job building buildings is akin to our sacrifice, you are attacking our sacrifice," their letter said.
"This goes beyond politics. It is about a sense of decency. That kind decency you mock as 'political correctness'," it went on to say.
Mr Trump's running mate, Governor Mike Pence, released a statement on Sunday avoiding the crossfire and saying he and Mr Trump both believed Captain Khan was an American hero.
However, Mr Trump took to Twitter on Monday to criticise the Khans for appearing on television, adding that the story was not about Khizr Khan, but rather "radical Islamic terrorism" and the US.
Mr Trump's former campaign manager Corey Lewandowski also defended his former boss, saying Captain Khan would still be alive if the billionaire was president "because he would've never engaged in a war that didn't directly benefit this country".
Democratic presidential candidate Hillary Clinton attacked Trump's treatment of the Khans during a campaign stop at a church in Cleveland, Ohio.
"Mr. Khan paid the ultimate sacrifice in his family, didn't he?" she told the African American congregation.
"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," she said.
Pew Research Center's 2014 Religious Landscape Study found that 0.9% of US adults identify as Muslims.
A 2011 survey of Muslim Americans, estimated that there were 1.8 million Muslim adults (and 2.75 million Muslims of all ages) in the country. That survey also found that a majority of US Muslims (63%) are immigrants.
Demographic projections estimate that Muslims will make up 2.1% of the US population by the year 2050, surpassing people who identify as Jewish on the basis of religion as the second-largest faith group in the country (not including people who say they have no religion).
A Pew Research Center report estimated that the Muslim share of immigrants granted permanent residency status (green cards) increased from about 5% in 1992 to roughly 10% in 2012, representing about 100,000 immigrants in that year.
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