The Hammers were granted arbitration after the FA rejected their appeal against the suspension for Carroll's dismissal in the 2-0 home win over Swansea on Saturday.
"The Independent Tribunal resolved that there was no serious issue to be tried and also awarded The FA its costs," the FA said in a statement.
Referee Howard Webb sent Carroll off after he clashed with Swansea defender Chico Flores in the teams' relegation tussle at Upton Park.
West Ham said Carroll caught Flores by accident with his arm and even threatened legal action after the initial appeal was rejected and could do so again.
West Ham are 18th in the standings, just inside the relegation zone, with 22 points from 24 matches while Swansea are two points above them in 12th and have sacked manager Michael Laudrup.
(Reporting by Rex Gowar, editing by Mark Meadows)
