Mr McCain said Mr Trump did not have an "unfettered licence to defame those who are the best among us".
Democratic lawmakers and the soldier's father have called on Republicans to disavow Mr Trump.
US Army Capt Humayun Khan was killed by a car bomb in 2004 in Iraq at the age of 27.
His father, Pakistani-born Khizr Khan, told the BBC on Monday that Mr Trump could not insult women, judges and even members of his own party and not expect to face criticism.
"We all have same equal rights," Mr Khan said.
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."
"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."
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.