McCain, the 2008 Republican presidential nominee, was asked by CNN's Manu Raju about Trump's tweet about flag burning.
"I have not been commenting on Mr. Trump and I will continue not to comment on Mr. Trump," McCain told Raju.
On Tuesday, Trump proposed a penalty -- including possible jail time or loss of citizenship -- for burning the American flag.
"Nobody should be allowed to burn the American flag -- if they do, there must be consequences -- perhaps loss of citizenship or year in jail!" he tweeted. The most recent national discussion on flag burning was prompted by protests stemming from the election results at Hampshire College in Massachusetts.
But McCain did provide his opinion on flag burning: He doesn't approve but said the Supreme Court ruled that people are "free to express themselves that way."
"I think there should be some punishment, but right now the Supreme Court decision is that people are free to express themselves that way. But I do not approve of it and I think there's other ways for people to express their views rather than burn a flag that so many Americans fought and died for," said the Vietnam War veteran and POW.
When asked about Trump suggesting people could lose their citizenship over burning a flag, McCain reiterated that he would not comment.
"I do not comment on Mr. Trump's comments. I am not and will not," he said. "Because that's my choice ... I was just reelected by the people of Arizona and during that campaign, I did not comment on Mr. Trump, I will continue to not because I have to defend this nation. I'm the chairman of the Armed Services Committee. We have all kinds of challenges. My time is devoted to trying to make sure this nation is secure. Not to comment on every comment of Mr. Trump's."
He also told a Huffington Post reporter
on Tuesday to "tell all your friends" that he wouldn't be commenting on Trump.
McCain and Trump had a rocky history during the 2016 election that began when Trump said McCain "was not a war hero" because he got captured.
Since then, the two have had a series of public back-and-forths that include Trump initially not endorsing McCain for reelection in the Senate, and McCain withdrawing his support for the then-2016 Republican nominee.