The Irons are unhappy that the England striker's ban - a consequence of being sent off against Swansea last week after an altercation with defender Chico Flores - was not overturned by the FA, and took the case to tribunal in the pursuit of what Allardyce claimed was "justice".
However, the independent panel, at which the FA was also represented, found "there was no serious issue to be tried" and as such rejected West Ham's submissions, meaning Carroll must serve out his punishment.
The striker, who has just returned from injury to help boost the Hammers battle for Barclays Premier League survival, will now also miss the visit of Norwich next Tuesday night and then the trip to Southampton on February 22.
Carroll took to Twitter to vent his disappointment at the FA's decision to turn down his appeal earlier this week, writing "DISGRACEFUL!!".
Allardyce accepts the 25-year-old will need some time to cool down.
"I am frightened of going near him. He is hugely frustrated and angry and disappointed in this situation. It is best to leave him well alone and let him get over it," he said.
West Ham had stressed they would not be seeking any further legal redress, amid reports the club's owners were ready to take the matter to the High Court.
A brief club statement read: "Whilst West Ham United are obviously disappointed at the decision, as we have made clear throughout, we respect the rules of football and shall abide by them."
Despite Friday's eventual outcome, Allardyce had felt the incident could go some way to changing future perception of such cases.
"Even if this goes against us, we may get them sat down to create a better situation for this scenario in future. That is the least we can hope," he had said earlier on Friday.
Allardyce believes the whole situation could have been avoided if referee Howard Webb had "done his job correctly" on the pitch.
"He (Flores) is crawling all over Andy's back, it's an obvious free-kick. He (Webb) in clear view of the incident, but doesn't give it. That is my biggest problem with Howard," Allardyce said.
"I wouldn't be stood here talking about Andy Carroll's red card if Howard had done his job correctly in that first moment."
Allardyce continued: "He (Webb) is a world-class referee.
"We feel really disappointed in the procedure that has happened.
"We feel disappointed by the decision in the first instance and since then Andy has been wronged and wronged and wronged all the way through.
"It is has not got to this point and this level.
"As a football club we don't want to be sat here doing what we are doing, but we feel it is necessary to do it."