Stephen B. Morton, Associated Press
Republican presidential hopeful Donald Trump greets supporters before he speaks at his South Carolina Campaign Kickoff Rally in Bluffton, S.C., Tuesday, July 21, 2015.

BLUFFTON, S.C. — Donald Trump pushed back ever harder Tuesday against Republicans fed up with his provocations, disclosing one opponent's cell number in a fiery speech and lowering flags to half-staff on his properties as part of an in-your-face escalation of the feud.

Fellow GOP presidential contender Sen. Lindsey Graham, whose 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 Arizona 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 McCain's defense.

In a speech to hundreds of supporters in Bluffton, South Carolina, on Tuesday, Trump kept on McCain, accusing him 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 the senator said the businessman's inflammatory remarks about Mexican immigrants had brought out the "crazies." And, he said Tuesday he would no longer respond to Trump's comments.

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 cell phone 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 also ordered the American flags on his U.S. properties to be lowered, an act he said was to honor the five service members killed in last week's shooting in Chattanooga, Tennessee. The symbolism served, too, to underscore his claim that he has been a stronger supporter of veterans than McCain, despite the senator's central work in passing laws that overhauled the Veterans Administration and strengthened programs against suicide by service members.

The back-and-forth is the latest in a series of showdowns between Trump and fellow GOP candidates frustrated by Trump's brash campaign, which has often overshadowed their own in recent weeks.

In his speech, Trump brushed off the criticism he's faced, both from political rivals and others angered by his comments. He said he's had business success even with countries he's criticized, such as China, and negativity doesn't affect him. He also said his wealth insulates him because he's not beholden to donors who might not like what he says.

"It turned out I'm much wealthier than people thought," Trump said. "Nobody has to give to me."

An editorial in The Des Moines Register, in early-voting Iowa, urged him to "pull the plug on his bloviating side show" and quit 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 paper said. The first debate, next month, will host the top 10 candidates in national polling, meaning Trump is almost certain to qualify to be on stage.

Trump called the editorial "sophomoric" and accused the paper of unfair treatment of his campaign.

The storm over Trump is proving to be fodder for Democrats.

Senate Democratic leader Harry Reid noted that while Trump's GOP White House rivals were nearly unanimous in denouncing his comments on McCain, they were more tentative in responding to his earlier criticisms of Mexican immigrants as "criminals" and "rapists."

"There is an ugly truth behind that silence, and it is this: When it comes to immigration policy, there is no meaningful difference between the Republican Party and Donald Trump," Reid said.

Still, many of Trump's backers in early voting states argue the response to his comments about McCain has been overblown.

"I think it's going to improve his viability to be honest with you," said Lou Gargiulo, one of Trump's county chairmen in New Hampshire. "The more Mr. Trump is being beaten on by people, obviously the better his polling numbers are."

Colvin reported from Newark, New Jersey. Associated Press writers Catherine Lucey in Des Moines, Bill Barrow in Atlanta, and Jonathan Lemire in New York contributed to this report.