BLUFFTON, S.C. -- Donald Trump pushed back ever harder Tuesday against Republicans fed up with his provocations, disclosing one rival presidential candidate's mobile number in a fiery speech that escalated his feuding with his own party.

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 defence.

In a speech to hundreds of supporters in South Carolina on Tuesday, Trump kept up that drumbeat, accusing McCain of being soft on illegal immigration.

"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.

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. Even many non-Trump supporters doubted the latest controversy would hurt his appeal among a slice of the electorate that is deeply frustrated with the Washington establishment and finds his unvarnished persona refreshing.

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."

Trump responded during his speech by calling Graham an "idiot" and a "lightweight." He then held up a piece of paper and read out the senator's cellphone number to the capacity crowd of 540 people and the TV audience.

"Give it a shot," Trump encouraged. "He won't fix anything, but I think he'll talk to you."

Graham's voice mailbox was full Tuesday afternoon, said spokeswoman Brittany Brammell. Graham tweeted later: "Probably getting a new phone. iPhone or Android?"

Christian Ferry, Graham's campaign manager said Trump "continues to show hourly that he is ill-prepared to be commander in chief."

Trump's back-and-forth with Graham is the latest in a series of showdowns between the celebrity known as "The Donald" and fellow Republican candidates.

While McCain may seem to some like an unusual target, the Arizona senator and former Republican presidential nominee is deeply unpopular among many conservatives. Trump supporters say McCain started the spat by calling several thousand Trump supporters who attended a recent campaign rally in Phoenix, Arizona, "crazies." Trump late Monday softened his tone, albeit slightly, in an interview with Fox News: "Certainly if there was a misunderstanding, I would totally take that back," he said before calling on McCain to apologize to the Arizona voters he insulted.

Much of Trump's appeal is based on his brand as a political outsider and say-anything style, and backers acknowledge that what drives his appeal is sometimes going to get him into trouble.

The Des Moines Register, Iowa's largest newspaper, published an editorial Monday night calling on Trump to drop out of the race. "Trump has proven himself not only unfit to hold office, but unfit to stand on the same stage as his Republican opponents," the Register said.