The budget airline has reported the matter to Essex Police after a video of the row was shared tens of thousands of times on Facebook.
Social media users have expressed outrage at the clip, which shows Ryanair staff failing to reprimand the customer or remove him from the plane.
The row took place on a flight from Barcelona to Stansted before take-off on Friday afternoon.
At the beginning of the footage, the 77-year-old woman’s daughter can be heard telling the man 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****** c***” 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.”
The passenger, who has not been named, then shouts “don’t talk to me in a foreign language you stupid ugly cow” before calling her an “ugly black b******”.
Staff on the flight seem to do little to silence the man during the footage, leaving a passenger sitting behind to try and deal with the situation.
A member of the cabin crew comes over at one point and says: “Don’t be so rude okay, you have to calm down.”
David Lawrence, 55, filmed the footage and can be heard calling for the passenger to be thrown off the flight.
He told Sky News: “I’m so disappointed in Ryanair.
“They’ve done nothing.
“They’ve reported it to the police only after the media explosion, it’s been two days.”
The elderly woman, named only as Mrs Gayle in the Huffington Post, can be heard telling the man he stinks and needs a wash.
Her daughter told the news site that if a black person had behaved in the same way they would have been thrown off the flight immediately.
She said that the man turned on her mother because she had arthritis, meaning it took her some time to move out of the way so he could reach his seat.
The daughter said she had taken her mother, 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.
She 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.
The Twitter account @StanceGrounded posted the video on Sunday and had 28,000 retweets by 2pm.
Karl Turner, Labour MP for East Hull and shadow attorney general, tweeted: “He should have been removed from the flight and handed over to the police.
“We assume the aircraft was in the UK. If it was he definitely committed criminal offences. I fully expect @Ryanair to confirm the situation if in fact it was their flight.”
MSP for Greenock and Inverclyde Stuart McMillan posted: “This is really appalling @Ryanair. You will know the identity of the man and the level of training given to your staff.”
Ryanair said the issue has been reported to police and they cannot comment any further.
Essex Police said in a statement: “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.
“We would encourage anyone with information to contact us on 101.
“You can also contact Crimestoppers 100% anonymously on 0800 555 111.”