Carroll was dismissed by referee Howard Webb in last Saturday's 2-0 victory over fellow strugglers Swansea City, after it was adjudged he swung an arm into the face of Chico Flores.
The Swansea defender went to ground and theatrically clutched his face, with replays suggesting Carroll's arm had brushed the top of his head.
That led to outrage from the West Ham hierarchy, with manager Sam Allardyce saying his side had been the victim of "injustice" at his pre-match press conference ahead of Saturday's encounter with Aston Villa.
A statement from the FA read: "An Independent Arbitration Tribunal convened under FA Rule K has today dismissed a legal challenge brought by West Ham United FC and Andy Carroll in relation to the red card received by Carroll in the match between West Ham United FC and Swansea City FC on 1 February 2014.
"The Independent Tribunal resolved that there was no serious issue to be tried and also awarded The FA its costs."
