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.