Fellow Republican presidential contender Sen Lindsey Graham, whose phone number Trump read to an audience, called him a "jackass," only to see floods of Trump supporters jam his phone line.
Trump is now at odds with much of the Republican establishment after a series of incendiary comments, topped by his weekend mocking of Sen. John McCain's experience as a tortured prisoner of war in Vietnam.
Since then the real estate developer and reality TV host has intensified his criticism of McCain and his record on veterans issues in the Senate even as politicians from both parties and veterans groups have rushed to his defense.
"He's totally about open borders and all this stuff," Trump said. McCain sparked Trump's temper last week when he said his inflammatory remarks about Mexican immigrants had brought out the "crazies." And, he said Tuesday he would no longer respond to Trump's comments.
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Republican leaders and other 2016 candidates have been frustrated by Trump's brash campaign, which has often overshadowed their own in recent weeks.
But many Trump backers in early-voting states argue the exchange is being overblown by the media and his opponents, who fear his sudden rise in early polls.
Graham, a McCain friend and one of the 16 notable Republicans running for the presidential nomination, betrayed the growing exasperation and anger of many in the party when he appeared earlier on "CBS This Morning."
"Don't be a jackass," Graham said. "Run for president. But don't be the world's biggest jackass."
He said Trump had "crossed the line with the American people" and predicted this would be "the beginning of the end with Donald Trump."
"Give it a shot," Trump encouraged. "He won't fix anything, but I think he'll talk to you.