The debate was shaken by the death of Supreme Court Justice Antonin Scalia hours before the candidates took the stage. Among the contenders, only Jeb Bush said Obama had "every right" to nominate a justice during his final year in office. The former Florida governor said the presidency must be a strong office - though he added that he didn't expect Obama to pick a candidate who could win consensus support.
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The five other candidates on the stage urged the Republican-led Senate to block any attempts by the president to get his third nominee on the court.
"It's up to Mitch McConnell and everybody else to stop it," Donald Trump said. "It's called delay, delay, delay."
A debate that began with a somber moment of silence for Scalia devolved quickly into fighting between Trump and Bush, then between Trump and Cruz. The exchanges highlighted the bad blood between the billionaire businessman and his rivals as the race turns to South Carolina, a state known for rough-and-tumble politics, where the next Republican primary will take place in one week.
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Trump, repeatedly interrupting his rivals, lashed out at Cruz after the Texas senator challenged his conservative credentials, calling him the "single-biggest liar" and a "nasty guy." The real estate mogul also accused Bush of lying about Trump's business record and said Bush's brother - former President George W. Bush - lied to the public about the Iraq war.
Bush, who has been among the most aggressive Republican candidates in taking on Trump, said that while he didn't mind the businessman criticizing him - "It's blood sport for him" - he was "sick and tired of him going after my family."
Trump was jeered lustily by the audience in Greenville, South Carolina, a state where the Bush family is popular with Republicans. George W. Bush plans to campaign with his brother in Charleston Monday, making his first public foray into the 2016 race.
Ohio Gov. John Kasich sought to inject the election's high stakes into the discussion in the midst of the fiery exchanges between his competitors.
"I think we're fixing to lose the election to Hillary Clinton if we don't stop this," Kasich said.
The governor's warnings did little to deter his feisty colleagues.
Cruz and Sen. Marco Rubio also revived their fight over immigration, with the Texas senator haranguing his Florida counterpart for sponsoring failed legislation that would have created a pathway to citizenship for many of those in the United States illegally. Cruz also accused Rubio of taking a more moderate approach when speaking to Spanish-language media in an attempt to appeal to Hispanics.
"I don't know how he knows what I said on Univision - he doesn't speak Spanish," Rubio shot back.
Rubio entered the debate under immense pressure following his disappointing fifth-place finish in the New Hampshire primary. He stumbled badly in a debate days before that vote.
Rubio appeared more fluid in Saturday's contest, including during a robust defense of his proposed 25 percent corporate tax rate - which is not as much of a tax cut as many of his rivals are pitching. Rubio said his idea would leave enough revenue in the federal budget to triple the child tax credit for working families with children.
Just six contenders took the debate stage, far from the long line of candidates who participated in earlier GOP events. Yet the Republican race remains deeply uncertain, with party elites still hoping that one of the more mainstream candidates will rise up to challenge Trump and Cruz. Many GOP leaders believe both would be unelectable in November.
Scalia's sudden death - and the chance to replace him - could serve as a reminder for voters of the consequences of elections.
Cruz cast the moment in stark terms, saying allowing another Obama nominee to be approved would amount to Republicans giving up control of the Supreme Court for a generation.
"One of the most important judgments for the men and women of South Carolina to make is who on this stage has the background, the principle, the character, the judgment and the strength of resolve to nominate and confirm principled constitutionalists to the court," Cruz said.
Retired neurosurgeon Ben Carson is fighting to stay in the mix in South Carolina. He was overshadowed in the debate by his more aggressive rivals but lined up with most of the field in saying he agreed Republicans should not allow a Supreme Court justice to be appointed during Obama's final year in office.
Bush and Kasich both see an opening in South Carolina after Rubio's stumbles.
Kasich defended himself against attacks on his conservative credentials, particularly his decision to expand Medicaid in Ohio despite resistance from his GOP-led Legislature. Kasich argued that his decision was a good deal for the state in the long run.
"We want everyone to rise and we will make them personally responsible for the help they get," said Kasich, whose fledgling campaign gained new life after a second-place finish in the New Hampshire primary.
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