Sen. Lindsey GrahamLindsey Olin GrahamTrump, Europe increasingly at odds on Iran Trump: Anonymous news sources are 'bulls---' Trump: 'Good chance' Dems give immigration 'win' after Pelosi called White House plan 'dead on arrival' MORE (S.C.) blasted fellow Republican presidential candidate Donald TrumpDonald John TrumpTrump: 'I will not let Iran have nuclear weapons' Rocket attack hits Baghdad's Green Zone amid escalating tensions: reports Buttigieg on Trump tweets: 'I don't care' MORE on Monday night, calling the businessman a "jackass" for his comments on Sen. John McCainJohn Sidney McCainPelosi receives John F. Kennedy Profile in Courage Award Romney: Trump 'has distanced himself from some of the best qualities of the human character' MSNBC host: Barr 'the most dangerous person' who works for Trump MORE's (R-Ariz.) war record.

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"What he said about John, I think, was offensive. He's becoming a jackass at a time when we need to have a serious debate about the future of the party and the country," Graham said on CNN’s "Erin Burnett OutFront."

"This is a line he's crossed, and this is the beginning of the end of Donald Trump. ... I am really pissed," Graham added.

Graham repeated his criticism of Trump on Tuesday morning, saying he didn't think there was "a market for slandering POWs" in the Republican Party.

"I don’t care if he drops out," Graham said on "CBS This Morning," after a top Iowa newspaper called on Trump to drop out of the race in the wake of his remarks.

"Stay in the race, just stop being a jackass. You don’t have to run for president and be the world’s biggest jackass — that’s not your choices," Graham said on CBS.

Trump caught widespread flak over the weekend for questioning McCain's status as a war hero, saying that he preferred those who did not get captured. He has since attempted to clarify those remarks.

McCain, who was held as a prisoner of war during Vietnam for nearly six years, brushed off the comments earlier Monday, saying Trump might owe an apology to other POWs.

Graham, a close friend of McCain's, is a long shot for the White House in an extremely crowded GOP field, but has sought to make national security a primary focus of the 2016 race.

Updated at 10:16 a.m.