Republican frontrunner Donald Trump has attacked Ted Cruz over his birth in Canada, saying it raises questions about his presidential eligibility.
In the latest Republican debate for White House hopefuls, Mr Trump told his rival: "There's a big question mark over your head."
The constitution mandates the president be a "natural born citizen" of the US.
Issues of national security, the economy and foreign policy have also played heavily in the debate.
In the polls, the pair are leading the five other candidates, who were also on the stage in North Charleston.
The debate came just two weeks before the first real test of the campaign, when voters in Iowa pick their Republican and Democratic choices for president.
US media verdict on the debate
"Cruz acquitted himself well, cementing his status as the front-runner's chief opponent," writes Howard Kurtz for Fox News. "But Trump didn't suffer, and in fact may have had his strongest debate performance... The two-man top tier remains just that, way ahead of the rest of the field."
"For much of his career in Washington, Ted Cruz has been dismissed as a cartoonish sideshow," Michael Barbaro notes in the New York Times. However, he "did not just dominate much of the Republican debate, he slashed, he mocked, he charmed and he outmanoeuvred everybody else on stage".
"Donald Trump and Ted Cruz had an unofficial non-aggression pact at the first five Republican presidential debates... but the sixth one Thursday night quickly became a flurry of mutual scorn," writes Susan Page in USA Today.
"There were only three real players in this exercise: Trump, Cruz and Rubio," according to Josh Marshall on the Talking Points Memo website. "Trump wins, Cruz loses a bit of ground but not much and the clock continues to run out on Rubio."
- Mr Cruz defended a controversy over his campaign finances in 2012, alleging bias in the media
- He said Mr Trump embodied "New York values", adding "not a lot of conservatives come out of Manhattan - I'm just saying"
- Mr Trump defended his call for a halt on Syrian refugees, saying they were a "Trojan horse" bringing in people who would harm the US
- He came under fire from Florida Governor Jeb Bush for proposing a ban on Muslims coming to the US
- Florida Senator Marco Rubio said Mr Obama would confiscate every gun in the US if he could
- The candidates bickered over how to respond to China's impact on the US economy
Analysis - Anthony Zurcher, BBC News, Washington
The sixth Republican presidential debate was the political equivalent of a wrestling "battle royale", where fists fly, chairs are thrown, the crowd cheers and the referees flee for safety.
There could have been no clearer indication that the Cruz-Trump honeymoon was over. The two candidates who stand atop the Republican presidential opinion polls had exchanged warm words through much of the campaign but with the Iowa caucuses just two weeks away, the niceties have melted away.
They exchanged barbs over Mr Cruz's Canadian birthplace and Mr Trump's alleged liberal "New York values".
From there, numerous side fights broke out. Rubio v Christie over Mr Christie's tenure as New Jersey governor. Trump v Kasich, Rubio v Bush on trade. Rubio v Cruz on immigration. Each candidate could boast a strong moment or two, but each also felt the sting of their competitors' barbs.
A battle royale is supposed to end when only one participant is left standing. In Charleston, however, all the candidates survived - but all were bloodied.
The event hosted by Fox Business Network came after days of Mr Cruz and Mr Trump taking shots at each other, shattering a months-long period of goodwill between the two men.
The start of hostilities began a week ago when the billionaire businessman started raising questions about whether the Texas senator's birth in Canada put his eligibility in doubt.
But on the debate stage on Thursday night, Mr Cruz said there was "zero chance" of a lawsuit succeeding because the constitution's definition of "natural born citizens" included people born to an American parent.
Mr Cruz was born in Calgary to an American mother and a Cuban father.
But the business mogul stood firm, noting that a Harvard law scholar had raised doubts and Mr Cruz could face lawsuits by Democrats wishing to challenge his qualification.
Candidates for US president must
- be a "natural born citizen" - this has never been tested in court but is widely interpreted as being a US citizen at birth, so born in the US or having a US citizen parent
- be 35 years of age or older
- have lived in the US for the past 14 years
They also argued over the meaning of "New York values", which Mr Cruz threw at the New York billionaire as a slur on his conservative credentials.
But the New Yorker said that was an insult to the "great people" who pulled together after the 9/11 attacks.
After the debate, Mr Trump told reporters: "I guess the bromance is over."
All the candidates targeted leading Democratic candidate Hillary Clinton, who was repeatedly attacked for her time as Secretary of State.
The primary contests, in which each party picks their nominee for president, begin in February and the presidential election is in November.