Greenville, South Carolina (CNN) The sudden death of Supreme Court Justice Antonin Scalia provided a rare moment of agreement among Republican presidential candidates at a debate that quickly fractured into the most bitter and vicious brawl of the 2016 cycle.

An event that began with a sober moment of silence to honor Scalia, whose death was announced on Saturday, turned abruptly accusatory as the six GOP candidates unleashed a torrent of attacks and personal name calling, including multiple charges of lying. The sparring played out one week ahead of the South Carolina Republican primary, an event that has become legendary over the years for its bare-knuckle politics.

The White House contenders opened the night with praise for Scalia. They largely agreed that his successor, who could shift the ideological balance of the court, should be nominated by the next president. Front-runner Donald Trump said he was certain Obama would make a nomination whether Republicans like it or not and called on Senate Majority Leader Mitch McConnell to slam the brakes.

"It's called delay, delay, delay," Trump said.

Sen. Patrick Leahy, the top Democrat on the Senate Judiciary Committee, will appear Sunday morning on CNN's "State of the Union" to discuss Scalia and his succession on the Supreme Court.

Ted Cruz, battling with Trump for the anti-establishment mantle, said Scalia's passing underscored the high stakes of the 2016 election. The country is only "one justice away," the Texas senator warned, of the Supreme Court striking down important decisions on issues like abortion and religious liberty.

John Kasich, fresh off a second-place finish in New Hampshire, and Jeb Bush, hoping for a strong finish in a state that both his father and brother won, agreed that the next administration should choose Scalia's successor. Bush said the President "of course" has the right to nominate a Supreme Court justice, but added he was certain Obama's pick would not have consensus support.

"The next President needs to appoint someone with a proven conservative record similar to Justice Scalia," the former Florida governor said.

Marco Rubio hailed Scalia as "one of the great justices of the history of this republic," praising his life-long efforts to defend the "original meaning of the Constitution."

But the consensus at the GOP debate hosted by CBS News was short-lived as the candidates spent the next two hours clashing on issues including immigration, foreign policy and the legacy of former President George W. Bush.

'Lies'

Rubio and Cruz got into a cage match over their records on immigration reform. When asked about his plan to curb illegal immigration, Cruz said there was "sharp difference on immigration" between the candidates on stage, before delivering his first punch at Rubio on the issue.

"When Harry Reid and Chuck Schumer and establishment Republicans were leading the fight to pass a massive amnesty plan," Cruz said, he was on the side to defeat what he called the "Rubio-Schumer amnesty plan."

Rubio accused his Senate colleague of spreading "lies."

"He either wasn't telling the truth then, or isn't telling the truth now," Rubio said. "To argue that he's a purist on immigration is just not truth."

When Cruz said Rubio has a long record of supporting amnesty and referenced comments he made on Univision, Rubio shot back: "I don't know how he knows what I said on Univision because he doesn't speak Spanish."

Cruz quickly responded by speaking in Spanish.

Rubio wasn't the only one to call Cruz a liar. Moments later, Trump called the Texas senator "the single biggest liar."

"This guy lied about Ben Carson when he took votes away from Ben Carson in Iowa," Trump said, a reference to Cruz allies incorrectly telling voters on the night of the Iowa caucuses that Carson was dropping out of the race.

Trump went on to accuse Cruz of running robo-calls in South Carolina informing voters that Trump was not running in this state.

"This is the same thing he did to Ben Carson. This guy will say anything — nasty guy," Trump fumed.

Cruz accused Trump of hypocrisy, noting that earlier in the campaign, the businessman had called Carson "pathological" and compared the retired neurosurgeon to a child molester.

Looking on at his rivals from the edge of the stage, Carson appeared amused by the vicious attacks — and said he had no intention of joining in the mud slinging.

"So many people have said to me, you need to scream and jump up and down like everybody else," Carson said.

More than once Tuesday night, Kasich also tried to play a similar role of happy warrior.

Voters "want to see unity and I don't want to get into all this fighting tonight," the Ohio governor said. People are "sick of the negative campaigning."

Clashing over George W. Bush

One of the most electric — and personally charged— exchanges of the night took place between Trump and Bush over the legacy of George W. Bush, who will hit the campaign trail in North Charleston on Monday to stump for his brother for the first time.

It would prove to be one of several strong moments for Bush Saturday night, in what was easily one of his strongest debate performance of the cycle.

Trump started the fight by calling the war that George W. Bush waged in Iraq a "big fat mistake."

Jeb Bush hit back strongly, saying, "I'm sick and tired of Barack Obama blaming my brother for all of the problems" he faces. He then turned the fire on Trump: "While Donald Trump was building a reality TV show, my brother was building a security apparatus" to keep the country safe, he said.

Trump lashed back, saying it was under Bush's watch that the World Trade Centers fell on September 11, 2001.

"That's not keeping us safe," the New Yorker said, spurring the audience at the Peace Center to boo loudly in disapproval of Trump.

Rubio joined in the conversation by coming to George W. Bush's defense.

"He kept us safe and I'm forever grateful," the Florida senator said. The terrorist attacks of 2001 happened, Rubio added, because Bill Clinton failed to kill Osama bin Laden when he had the chance.

Trump's line of attack was clearly intended to remind voters of Bush's family legacy and revive hesitations about installing a political dynasty in the White House. But South Carolina is home to a large number of military personnel and veterans and Trump's critique of the Iraq War could prove risky here.

What's at stake

For Bush, the quest to win South Carolina is personal. The former governor is looking to capitalize on George W. Bush's popularity in the state by bringing the former president out to campaign with him this week. Lindsey Graham, the South Carolina senator who ended his own bid for president in December and endorsed Bush, played the role of surrogate Saturday night as he mingled with reporters in the spin room.

Rubio also has reason to hope for a strong showing here. Some of his top campaign aides, including campaign manager Terry Sullivan, are Palmetto State veterans who have been laying the groundwork in the state for months. The Florida senator also earned the endorsement of two prominent members of Congress from this state: Sen. Tim Scott and Rep. Trey Gowdy.

The pressure was intense for Rubio, who was rattled by former rival Chris Christie during a memorable moment at the last debate in New Hampshire. But he avoided any gaffes Saturday, and appeared nimble on the debate stage.

For Kasich, South Carolina is an uphill battle. The Ohio governor has largely campaigned on a moderate message, and though he is riding high from a second-place finish in New Hampshire this week, appealing to a more conservative base in South Carolina will be a tough task.

Carson's path forward is unclear. He briefly topped national polls and was competitive in Iowa, where he appealed to the state's large evangelical constituency. But after a series of missteps and the loss of top campaign staffers, Carson has not been able to regain momentum.

As he has in the previous debates, Carson on Saturday used self-deprecating humor to acknowledge that his moment in the spotlight appears to have passed.

"Thank you for including me in the debate," he said. "Two questions already -- this is great."