Host Tucker Carlson showed the video on Monday night, arguing it "does not show an insurrection or a riot in progress", but rather "mostly peaceful chaos".

A top congressional Republican gave Mr Carlson exclusive access to the video.

But some Republicans say the Fox host mischaracterised the raid on Congress.

The top-rated host on Fox News, a conservative network, has long insisted that other media outlets exaggerated violence at the Capitol on 6 January 2021, when Trump supporters stormed the complex as lawmakers met to certify Joe Biden's victory in the presidential election. He has also suggested without evidence that government agents could have instigated the riot.

In the roughly 45-minute segment, Mr Carlson said the video showed that while a minority of protesters did commit violence, most were "sightseers".

His show on Tuesday night included an interview with Tarik Johnson, a former Capitol Police officer who said he donned a pro-Trump red hat during the riot in order to help fellow officers escape the melee.

Mr Carlson questioned why intelligence about possible violence that day wasn't relayed to rank-and-file officers, but he did not broadcast much new video from the riot. 

Criticism of Monday's segment came from Democrats as well as top Republicans in Washington, the head of the Capitol police, and the family of a police officer whose death was mentioned by Mr Carlson in the show.

Senate Republican leader Mitch McConnell said on Tuesday it had been a "mistake for Fox News to depict this in a way that's completely at variance with what our chief law enforcement official here at the Capitol thinks" about the riot.

Mr McConnell pointed to an internal memo by Capitol Police Chief Tom Manger, whose agency is responsible for protecting the buildings where the lawmakers meet.

In that memo, Mr Manger says the primetime Monday broadcast was "filled with offensive and misleading conclusions about the January 6 attack".

"The programme conveniently cherry-picked from the calmer moments of our 41,000 hours of video," he wrote.

"The commentary fails to provide context about the chaos and violence that happened before or during these less tense moments."

During Monday's programme, the Fox host showed footage of the "QAnon Shaman" - a man named Jacob Chansley - wandering around the Capitol building trailed by police officers who show no signs, at least in the clips broadcast, of attempting to stop or arrest him.

Chansley, who was bare-chested and wearing a headdress made of fur and horns that day, became one of the most recognisable figures from the Capitol riot.

He pleaded guilty in September 2021 to obstructing an official proceeding and is currently serving a 41-month prison sentence.

Mr Carlson said the videos - which have not been viewed by the BBC or any other media organisation in full - showed that police had acted as Chansley's "tour guide".

This version of events was rejected by Mr Manger, who called the allegation "outrageous and false".

"Those officers did their best to use de-escalation tactics to try to talk rioters into getting each other to leave the building," he wrote.

The justice department says about 140 police officers were assaulted that day.

Around 1,000 people have been arrested so far in connection with the riot, most of them charged with entering or remaining in a restricted federal building, according to justice department figures.

More than 300 have been charged with assaulting, resisting, or impeding officers or employees, including more than 100 who have been charged with using a deadly or dangerous weapon or causing serious bodily injury to a police officer.

Mr Carlson also focused on Capitol Police officer Brian Sicknick, who collapsed after returning to his office during the siege and died the next day.

Fox News showed footage of Mr Sicknick continuing to perform his duties inside the Capitol after being pepper-sprayed.

"Whatever happened to Brian Sicknick was very obviously not to do with the violence he suffered outside the Capitol," Mr Carlson said.

In April 2021, a medical examiner found that Mr Sicknick had died of natural causes from a medical condition not brought on by an injury.

On Tuesday Mr Sicknick's family issued a statement saying they were "outraged" at the coverage and lambasted the network as "propaganda".

Meanwhile, there were new revelations on Tuesday night from a $1.6bn (£1.2bn) defamation case filed by Dominion Voting Systems against Fox News. The electronic voting firm accuses the network of promoting baseless claims of vote-rigging.

The latest legal filings in the case reportedly suggest that two days before the Capitol riot Mr Carlson wrote in text about his "hate" for Mr Trump,

"We are very, very close to being able to ignore Trump most nights," he wrote on 4 January 2021, according to US media. "I truly can't wait."

"I hate him passionately," he added.

Reports of the filing emerged as Mr Trump himself posted on his social media platform Truth Social in all capital letters: "Great job by Tucker Carlson tonight."