McLean must also undertake an education program and has been given a suspended one-match ban.
The 26-year-old initially posted "Today can f*** right off" on Twitter on Tuesday night, which prompted another user to ask if he had been delisted.
McLean, who has struggled to make an impact since joining Carlton at the end of the 2009 season, replied with a comment saying he'd contracted AIDS from the user's mother.
McLean's post came just one week after the Blues were fined $7500 after three of their players used Twitter to criticise umpires.
McLean said he had been upset to receive criticism from an anonymous football fan, but that was no excuse for his crude response.
"I realised soon after that my comments were not suitable and took it down and I sincerely apologise for offending anyone who read the tweet in the short time it was up," he said in a statement announcing his penalty on Wednesday night.
"The discussions with the club this morning were an important reminder of the possible ramifications of comments in social media and of remembering the added responsibilities that come with being a player with the Carlton Football Club."
Despite the Blues' recent Twitter troubles, football manager Andrew McKay said players were encouraged to be involved with social media, but McLean's mistake was an important reminder of potential pitfalls.
"We believe it brings the supporters closer to the club and in particular the players," McKay said.
"We continue to work with the players in regards to what is acceptable and we are confident that Brock will not transgress again."