The man was filmed by a fellow passenger calling the elderly woman an "ugly black b******", and shouting "don't talk to me in a foreign language you stupid ugly cow".
-> Two dead in horror smashes on same Yorkshire roadThe footage was recorded on October 19 on Ryanair flight FR015 from Barcelona to London Stansted by Kent resident David Lawrence who uploaded it on to social media.
It shows the man shouting at the woman to move seats while her daughter tries to stand up to him, telling him her mother is disabled.
He replies: "I don't care whether she's f****** disabled or not - if I tell her to get out she gets out." The man then calls her "an ugly f****** ****" and tells staff to move the woman to another seat, adding: "if you don't go to another seat I'll push you to another seat".
Staff on the flight seemed to do little to silence the man during the footage, leaving a passenger in the row behind to try and deal with the situation.
The cabin crew can be heard to say: "Don't be so rude, you have to calm down" while other passengers call for the man to be thrown off the flight.
The daughter of the woman who was targeted said she believed if a black person had behaved in the same way they would have been thrown off the flight immediately.
The woman, 53, told the Huffington Post the row had started because her 77-year-old mother's arthritis meant it took her some time to move out of the way so the man could reach the window seat.
The daughter said she had taken her mother, who is a Windrush Generation migrant who came to the UK from Jamaica in the 1960s, on holiday to mark one year since the death of her father.
"My parents were married over 50 years. She's been feeling really down and depressed, so I thought the trip would raise her spirits." She continued: "I know that if I was behaving like he was or any other black person for that matter, police would have been called and we would have been kicked off the flight."
"Mum's really feeling upset and very stressed about this situation, on top of the grief she's already experiencing. As for me, I'm upset about the whole thing too - the fact that the passenger wasn't taken off the plane and how the situation was dealt with," she said.
The woman said that when she had complained to cabin crew they denied hearing any racial slurs and they told her to ring customer services on Monday.
News of the incident was retweeted nearly 40,000 times after a user with the handle @StanceGrounded uploaded Mr Lawrence's clip. Scores of passengers have now declared a boycott against the Dublin-based budget airline for their apparent lack of action.
Ryanair tweeted on Sunday that it had seen the footage and reported the incident to Essex police. It added: "As this is now a police matter, we cannot comment further."
A spokeswoman for Essex police said: "This incident, which we were made aware of this morning, is believed to have taken place on a plane at Barcelona Airport. "Essex Police takes prejudice-based crime seriously and we want all incidents to be reported. "We are working closely with Ryanair and the Spanish authorities on the investigation."
Why the man who filmed the incident has refused to apologise
A man who filmed a racist incident on a Ryanair flight said he chose not to intervene so he could publicise what happened.
David Lawrence, who filmed a man subjecting a woman to a tirade of racist abuse, said that in the age of social media "visual is important" and "now the world knows" what happened.
He told BBC Breakfast: "I had to make a difficult decision at that time because if I had stepped in I don't think you would have seen the footage that I captured."
Prime Minister Theresa May's official spokesman declined to comment on the specific case, but told reporters: "When people are travelling and going about their public life, no-one should be subjected to intimidation or any form of abuse.
"The Prime Minister has always been clear that racial abuse is abhorrent.
"In relation to this particular case, there's obviously an investigation being conducted by Essex Police and it's right that I don't comment specifically on that."
The video, recorded on October 19 on flight FR015 from Barcelona to London Stansted before the plane had taken off, shows a man shouting at a woman telling her to move seats while her daughter tries to stand up to him, telling him her mother is disabled.
He said: "I don't care whether she's f****** disabled or not - if I tell her to get out she gets out."
In the video the man can be heard calling the elderly woman an "ugly black b******", and shouting "don't talk to me in a foreign language you stupid ugly cow".
He was not confronted by any other passengers, Mr Lawrence added.
"In fact, one of the passengers whilst I was filming basically said to me don't you think you're being a bit childish by filming this," he said.
Ally Murphy, a former cabin crew supervisor who also appeared on the show, said Mr Lawrence made the right call in deciding to film the incident.
"It is great that David did film that," she said. "This happens every day in the community and people need to step forward and say this isn't right."
"But as a cabin crew member it is not great when people pile in and try to take things into their own hands," she added.