Chelsea Football Club said it took the "strongest action" against Fitzwater, saying it "abhors" discrimination.
Fitzwater admitted racially aggravated abuse at the court hearing and he was also given a three-year football banning order preventing him from attending any regulated football match in the UK or abroad.
A Chelsea spokesman said: "Chelsea Football Club and the overwhelming majority of our fans abhor all forms of discrimination and believe they have no place in our club or our communities.
"We would like to thank Chelsea fans for reporting this incident and others which occurred during the semi-final, as a result of which a number of persons have been banned for using racial and abusive language."