The scene unfolded after Davis said he and his family boarded a United Airlines flight from Denver to Orange County on Saturday.

The two-time Super Bowl champion said in an Instagram post on Monday that his son had asked a flight attendant for a cup of ice during drink service ... but when the steward didn't respond to the request, Davis says he leaned over and calmly tapped the man's arm to get his attention.

Davis, though, says the guy snapped ... and shouted "don't hit me," before he bolted from the area.

Davis says he had no further interactions with the man for the duration of the flight -- but once the plane touched down in California, "the pilot asked all passengers to remain seated" -- and six law enforcement agents, including ones from the FBI, hopped on board and immediately placed him in cuffs.

"I was then removed from the flight while being recorded by multiple passengers as I was paraded off of the plane in handcuffs by the officers," Davis said.

A short time later, however, Davis says he was cleared of any wrongdoing ... writing on IG, "the agents profusely apologized" and even offered "to support me and my family in any way possible."

Davis is now demanding an explanation from United Airlines ... and he seemed to be threatening to hit the company with a lawsuit over it all -- alleging the flight attendant caused him potentially irreparable harm with false allegations.

"These lies and the ensuing events have the capacity to tarnish the reputation I have spent decades building," Davis said, "and this person should not be able to do this again to anyone in the future."

For the FBI's part, it said in a statement to TMZ Sports on Sunday its agents did respond to John Wayne Airport over a report of an incident on a plane ... and did detain an individual for further questioning.

The FBI did not identify Davis by name ... but did add that the person "was cooperative with law enforcement and was released to continue his travels."