President Trump threatened Arizona Sen. John McCain, a former POW with a terminal brain cancer diagnosis, over his comments at an awards ceremony Monday night which were critical of U.S. leadership. Mr. Trump was asked whether he had heard McCain's remarks.

"People have to be careful because at some point I fight back," Mr. Trump told WMAL radio talk show host Chris Plante in an interview Tuesday morning. "You know, I'm being very nice. I'm being very, very nice. But at some point I fight back and it won't be pretty."

McCain told CBS News' Alan He that he doesn't comment on what the president says. "I comment on what he does," he said. "And I will say I have faced some pretty tough adversaries in the past. I'm not interested in confronting the president, I'm interested in working with the president."

Get Breaking News Delivered to Your Inbox

The Arizona senator, who on Monday received the Liberty Medal for his lifetime of service and sacrifice, delivered a harsh critique of the direction the country has taken, although he did not mention the president's name in his remarks.

"To abandon the ideals we have advanced around the globe, to refuse the obligations of international leadership for the sake of some half-baked, spurious nationalism cooked up by people who would rather find scapegoats than solve problems," he said in accepting the award, "is as unpatriotic as an attachment to any other tired dogma of the past that Americans consigned to the ash heap of history."

McCain also urged the U.S. toward engagement with the world, rather than withdrawal and isolationism.

"We have done great good in the world. That leadership has had its costs, but we have become incomparably powerful and wealthy as we did. We have a moral obligation to continue in our just cause, and we would bring more than shame on ourselves if we don't," he argued. "We will not thrive in a world where our leadership and ideals are absent. We wouldn't deserve to."

CBS News' Arden Farhi contributed to this report.