The son of controversial club owner Massimo Cellino became involved in an abusive row with a supporter on Facebook.
The FA said Cellino's charge included an "aggravated breach" as his comments included a reference to disability.
Cellino subsequently apologised for his comments and denied the charge, but he was handed a suspension, which starts with immediate effect, following an Independent Regulatory Commission hearing on Monday.
He has also been fined £5,000, warned about to his future conduct, and must complete an education course within four months.