McCain, while accepting the Liberty Medal in Philadelphia on Monday night
, warned the United States against turning toward "half-baked, spurious nationalism cooked up by people who would rather find scapegoats than solve problems."
The speech was a repudiation of Trump, who the Arizona Republican has long feuded with, and the worldview that catapulted him to office.
Trump told Chris Plante of "The Chris Plante Show" on Tuesday that he heard the criticism and warned McCain to be careful.
"He was taking shots at you again yesterday," Plante said. "You heard what he said yesterday, Sen. McCain?"
"Yeah, well I hear it. And people have to be careful because at some point I fight back," Trump said. "I'm being very nice. I'm being very, very nice. But at some point I fight back, and it won't be pretty."
Responding to Trump's threat, McCain bluntly told reporters Tuesday: "I've faced far greater challenges than this."
McCain and Trump's feud has dated back years, ever since the then-2016 candidate said the senator wasn't a war hero because he was captured in Vietnam.
"He is not a war hero," Trump told pollster Frank Luntz in 2015.
"He is a war hero," Luntz interjected.
"He is a war hero because he was captured," Trump said, cutting him off. "I like people that weren't captured, OK? I hate to tell you. He is a war hero because he was captured."
The feud continued throughout the 2016 campaign, with McCain regularly faulting the Republican standard bearer for comments he made on the campaign trail. McCain withdrew his support for Trump when the "Access Hollywood" video was made public, showing the 2016 Republican nominee bragging about sexually assaulting women.
Both Trump and McCain won re-election, but their feud continued, with the Arizona senator casting a deciding vote against the Republican health care plan.
The act of defiance stung Trump.
"Sen. McCain, you mean the one who voted against Obamacare?" Trump asked, rhetorically, during a sweeping August news conference. "You mean Sen. McCain who voted against us getting good health care?"
This public back-and-forth contributed to McCain's speech Monday, where he was honored by Democrats and Republicans alike for a lifetime of service. McCain was diagnosed with brain cancer earlier this year.
The Arizona senator and his party's 2008 presidential nominee described that "half-baked, spurious nationalism" as "unpatriotic as an attachment to any other tired dogma of the past that Americans consigned to the ash heap of history."