Republican presidential candidate Donald Trump has criticised the three moderators of Thursday night's debate, particularly Fox News host Megyn Kelly.
He tweeted that Ms Kelly - who challenged the tycoon about his views on women - and her co-moderators were "not very good or professional".
The punchy debate was watched by 24 million people, a record for a primary contest, according to Nielsen.
Mr Trump was one of 10 candidates taking part in Cleveland.
They had been selected from a crowded field of 17 candidates by Fox News on the basis of recent national polls.
Mr Trump stumbled on his past support for a national healthcare system but his most uncomfortable moment came when Ms Kelly challenged him on his views about women.
"You've called women you don't like fat pigs, dogs, slobs and disgusting animals," she said.
He answered by joking that he only said that about actress Rosie O'Donnell and stating that political correctness was one of the country's biggest problems.
"I don't have time for total political correctness," he said.
In the hours after the debate, he took to Twitter to criticise the moderators.
And he also took aim at Republican pollster Frank Luntz, whose focus group in a live broadcast after the debate gave an overwhelming thumbs-down to Mr Trump's performance.
It was when Mr Trump said, during the debate, that he would not rule out running as an independent that the audience and other candidates became hostile. An enraged Rand Paul said: "He buys and sells politicians of all stripes."
One of the loudest rounds of applause of the evening was for Florida senator Marco Rubio when he mocked Hillary Clinton, who leads the Democratic field.
"First let me say, I think God has blessed us. He's blessed the Republican Party with some very good candidates. The Democrats can't even find one."
Analysis - Jon Sopel, BBC News, Cleveland
The received wisdom of political strategists was that last night was the opportunity for Donald Trump to look presidential - but the secret of his success is to look like, well, Donald Trump. There are many things that make him stand out from the rest of the Republican field but surely the most striking is that he is already a brand in his own right.
What's propelled him to the front of the Republican field is that he isn't just another politician. He taps brilliantly into America's disillusionment and anger with conventional Washington politics. He's made billions from building hotels, golf courses and casinos, so doesn't need to go cap in hand to get funding. He has instant name recognition across America as well through presenting the US version of the Apprentice, and he is a one man self publicity machine.
All of which leaves Republican establishment with a problem. If they do say you're fired, well then there's every possibility that Trump would run as an independent, draining support away from the conservative cause. Politics with him around could never be described as dull. But it is unpredictable. And you sense that all the Republican leadership want at the moment is a bit of certainty...even if that means doing without the undoubted charisma of Mr Trump.
On Thursday, the Democratic Party announced it would hold its first debate in Nevada in October, hosted by CNN.
By next summer, each party will have a presidential nominee who will do battle in the race for the White House. Votes will finally be cast in November 2016.
The Republican field is one of the largest in recent years.