McCain, who made the remarks while discussing the Vietnam War in an interview with C-SPAN3, echoed questions that surfaced during the 2016 campaign about Trump's medical history and draft deferments.
"One aspect of the [Vietnam] conflict, by the way, that I will never, ever countenance is that we drafted the lowest-income level of America, and the highest-income level found a doctor that would say that they had a bone spur," McCain said. "That is wrong. That is wrong. If we are going to ask every American to serve, every American should serve."
Though a spokeswoman for McCain, a sixth-term senator, told ABC News that he was not referring to the president, Trump was exempted from military service in the 1960s after receiving a letter from a doctor saying he had a bone spur ailment.
"I asked for student deferment, like many other people during the war or around the time of the war. I had a minor medical deferment for feet, for a bone spur of the foot, which was minor," Trump said in an interview with ABC News in July 2015. "I was not a fan of the Vietnam War. But I was entered into the draft, and I got a very, very high draft number."
McCain addressed the comments during an appearance on ABC's "The View" Monday, explaining that his critique was meant to address the injustice in American society at the time that allowed wealthy individuals to find a way out of service.
"What I was trying to say is one of the great inequities of the Vietnam conflict was the lowest-income Americans went and fought and were drafted and those where were wealthy enough to have a doctor to say, 'Hey, you got a bone spur or you got migraines' or whatever it is, then they were excused," said McCain.
"I think that when we ask the lowest-income portion of our public to do our fighting and dying for us, that that's disgraceful," he added. "Nothing makes me more angry than that."
His mention of bone spurs came after he and Trump exchanged words last week over the McCain's recent criticism of "half-baked, spurious nationalism."
"To fear the world we have organized and led for three-quarters of a century, to abandon the ideals we have advanced around the globe, to refuse the obligations of international leadership and our duty to remain the last best hope of earth for the sake of some half-baked, spurious nationalism cooked up by people who would rather find scapegoats than solve problems is as unpatriotic as an attachment to any other tired dogma of the past that Americans consigned to the ash heap of history," McCain said in an impassioned speech in Philadelphia last Monday.
When asked about the speech the following day, Trump said, "People have to be careful, because at some point I fight back."
"I'm being very nice. I'm being very, very nice. But at some point I fight back, and it won't be pretty," Trump told WMAL radio host Chris Plante last Tuesday.
McCain, who was tortured as a prisoner of war after his plane was shot down in Vietnam, downplayed the prospect of a feud with the president, telling CNN, "I've faced far greater challenges than this."
Trump has had strained relationship with McCain over the years. He sparked a public debate in 2015 when he told the Family Leadership Summit in Ames, Iowa, that McCain was "not a war hero" because he was captured.
"I like people who weren't captured," Trump said.
ABC News' Mariam Khan and Adam Kelsey contributed to this report.