Joseph Cassano, 23, quit his job as an emergency medical technician after several tweets became public that suggested he hated Jews, blacks and the poor, The New York Times reported.
His father, Salvatore J. Cassano, had yet to comment about his son's messages or resignation.
Mayor Michae Bloomberg, at a news conference Monday, said he had found the messages "offensive," and that the younger Cassano's resignation was "the appropriate thing."
Among the messages posted by Cassano, he said "I like Jews about as much as hitler #toofar? NOPE."
He also wrote "Getting sick of picking all these Obama lovers and taking them to the hospital because their medicare pays for an ambulance and not a cab," the New York Post reported.
In early March, Cassano wrote "my job is the worst."
Joseph Cassano and two other children of FDNY officials were among about 100 people recently hired by the city's emergency medical services department.
EMS workers have priority to be hired as firefighters.
Earlier Monday, Joseph Cassano apologized for the tweets.
"I regret posting some comments that were offensive, especially since I enjoy my job and treat every patient with great care and respect," he said in a statement released by the fire department.