Carroll was sent off after being deemed to have struck Chico Flores with his forearm and the Hammers took the matter to an independent tribunal on Friday after their appeal was turned down.
But their hopes justice would be served were again rejected, meaning Carroll has to serve his three-match ban, which began with their 2-0 win over Aston Villa over the weekend.
That still rankles with Nolan, who scored both goals at Villa Park, and he believes the FA was wrong not to overturn the original decision made by referee Howard Webb.
"We have nothing against the FA, they've just got to get it right," said Nolan, who admitted the injustice the Hammers are feeling fuelled their performance on Saturday.
"When people get things wrong, we expect them to hold up their hands and say, 'yeah, I got it wrong' and move on, like we all have to do.
"We have to hold our hands up as footballers when we get it wrong and take the stick that comes with it. I think they had a chance to redeem themselves and they haven't taken it.
"We were disappointed with the outcome. We thought it was an injustice. Andy is also frustrated, he's a good mate of mine and I felt really sorry for him.
"We've got to forget about Andy for the next couple of weeks now. He has to work hard and get himself ready because if we keep winning games it's going to be hard for him to get back into the team."