Republican presidential candidate Donald Trump on Monday offered rival Ted Cruz an ultimatum, threatening to sue Cruz over his eligibility to serve in the White House unless the Texas senator stops airing what Trump calls "false ads" and apologizes for what the billionaire real estate mogul called a series of lies about his positions.

With less than a week to go before South Carolina's pivotal Republican primary, the front-runner also reiterated that the 9/11 attacks happened during President George W. Bush's time in office — an apparent attempt to overshadow the former president's Monday campaign appearances on behalf of his brother, former Florida Gov. Jeb Bush.

The new attacks came as the race entered an increasingly nasty phase, with numerous negative ads airing on local television following an unusually caustic debate this past weekend.

Some of the harshest ads have been aimed at Trump, often using the political newcomer's past words to illustrate his evolving position on issues including abortion and gun rights.

Trump also took aim Monday at the Republican establishment, accusing the Republican National Committee of packing its debate audiences with donors — a move he claimed violated the loyalty pledge he signed in September vowing to run as a Republican and support the party's eventual nominee.

"I signed a pledge, but it's a double-edged pledge," Trump said at a luncheon in Mount Pleasant, South Carolina. "The pledge isn't being honored by the RNC."

RNC spokesman Sean Spicer responded to the criticism by saying: "Ten tickets from the RNC went to donors — 10."

But Trump saved the bulk of his criticism for Cruz. "If he doesn't take down his false ads and retract his lies," Trump said in a statement bashing Cruz, he will immediately file a lawsuit challenging Cruz's eligibility to serve as president.

Trump has previously said a federal court should decide whether Cruz meets the constitutional requirement of being a "natural-born citizen" to serve as president. Cruz was born in Canada to an American mother, and many legal experts have said he meets that test.

"Ted Cruz is a totally unstable individual. He is the single biggest liar I've ever come across, in politics or otherwise, and I have seen some of the best of them. His statements are totally untrue and completely outrageous," Trump said in the statement.

At events in the Charleston area and at a rally in Greenville, Trump piled on, slamming Cruz as "the most dishonest guy I think I've ever met in politics, "nuts" and a "basket case." He also questioned Cruz's faith.

"He goes around saying he's a Christian. I don't know, you're going to have to really study that," Trump said at a freewheeling news conference.

Cruz responded to Trump during a question-and-answer in Camden, characterizing Trump as nervous about his standing in the state.

"Today Donald Trump held a press conference. He apparently lost it. I mean, he was just going on and on about how I was the most horrible person in the world because I keep repeating the things he said," Cruz said to laughs. "And it's an amazing thing. Have you noticed how rattled Donald gets when his numbers start going down? He gets very, very upset.

"But I guess the only explanation one can have is his internal poll numbers in South Carolina must be plummeting following the debate," Cruz said.

Earlier, during an afternoon news conference, Trump also renewed his criticism of former President George W. Bush. While Trump would not say whether he considered Bush to be a failed president and declined to label him responsible for the 9/11 attacks, the former reality TV star repeatedly noted that "the Word Trade Center came down during the reign of George Bush."

Trump said his decision to go after George W. Bush — who made his campaign debut on his brother's behalf on Monday in South Carolina — was akin to his attacks on former President Bill Clinton, which began once the former president ramped up his campaigning for his wife, Democratic candidate Hillary Clinton.

"If the ex-president is campaigning for his brother, I think he's probably open to great scrutiny, maybe things that haven't been thought of in the past," Trump told reporters.

Trump said the real purpose of his news conference was to protest the Obama administration's consideration of the Consolidated Naval Brig in Charleston as a potential site for relocating detainees now housed at the U.S. Navy base in Guantanamo Bay, Cuba. Trump vowed to keep the Guantanamo prison open and stop the movement of any detainees to American soil.