"F*** the media", he shouts during the incident.
"Fortunately our cameraman is fine, he is made of stern stuff", O'Donoghue told the "Today" program on BBC Radio 4.
Another video of the altercation showed the cameraman resetting his video camera as the man is dragged away by what looks like another Trump supporter.
Ms. Montague said the president had been criticizing the media before the incident occurred.
The president had gone to El Paso to campaign for his border wall, an issue which resulted in the longest government shutdown in US history.
Nonprofit organization Reporters Without Borders joined in on the outrage, tweeting to its followers that it's "deeply concerned by this physical attack". The BBC team was later told by a campaign official that the "attacker was drunk".More news: Elizabeth Warren: Trump ‘May Not Even Be A Free Person’ In 2020
More news: Speaker Pelosi admonishes freshman Omar over Israel tweet
More news: Newborn baby rescued from storm drain in South Africa
"By the way, never wrestle him", Trump said of Gianforte at a campaign rally.
Former White House communications director Anthony Scaramucci condemned Trump's supporters at the Florida rally on Twitter, saying the behaviour was "not who we are".
Trump has frequently attacked the news media for what he views as unfair coverage, decrying some outlets as peddlers of "fake news" and "enemy of the people".
"I didn't know what was going on", Skeans said, according to the BBC, describing the moment when his camera suddenly skewed down and away from the stage.
United Nations experts last August warned that Trump's attacks "increase the risk of journalists being attacked with violence" and suggested that his rhetoric was "strategic".
"Ron is fine", the BBC says. She said the crowd had been whipped up into a frenzy by the president, and she shared video of the immediate aftermath of the attack, where she said her cameraman, Ron Skeans, was "assaulted".
Mr O'Rourke, meanwhile, held a countermarch with dozens of local civic, human rights and Hispanic groups in his hometown, followed by a protest rally attended by thousands on a baseball field within shouting distance from the arena where Mr Trump spoke.