Dramatic footage posted to social media showed two men brawling as a female flight attendant attempted to step in and passengers shouted at them to stop.
The airline confirmed officers in Tenerife had been called to meet the plane as it landed from Glasgow Prestwick on Saturday.
A statement from the budget firm said “after two passengers became disruptive in-flight... police removed and detained two individuals”.
Ben Wardrop, who filmed the incident, said it stemmed from a discussion which happened while the flight was still under way.
He told the Press Association: “It all started over a woman not wearing her shoes on the flight going to the toilet, and a very drunk man pulled her up about it, saying that someone would stand on her toes, then the woman’s boyfriend stepped in and tried to defuse the situation.
“Once the plane landed and more alcohol was consumed the man and the woman’s boyfriend started arguing, both being very drunk they started to fight, and the man in the video with lots of blood was trying to get the man to calm down and he got hit on the nose.
“One of the cabin crew, which was a young woman aged about 24, was caught in the middle of it all and was very scared, but done a good job on defusing the men fighting.
“There was a family with a young boy sitting in front of us that witnessed all of the action.
“It was a very scary moment as it was very confined and lots of innocent people got involved due to the small space on the plane.
“Ryanair’s staff handled the situation very well,” he said.
A photograph posted on Twitter appeared to show blood still on the door of an overhead luggage compartment before the plane’s next flight.
A Ryanair spokeswoman said: “The crew of this flight from Glasgow Prestwick to Tenerife (16 Mar) requested police assistance upon arrival after two passengers became disruptive in-flight.
“The aircraft landed normally and police removed and detained two individuals.
“We will not tolerate unruly or disruptive behaviour at any time and the safety and comfort of our customers, crew and aircraft is our number one priority.
“This is now a matter for local police.”
Tenerife police have been contacted for comment.