Riot police used tear gas to prevent activists distributing food from a lorry on Syntagma Square, where the country's parliament is located.
Athens Mayor Giorgos Kaminis says a Golden Dawn MP later tried to assault him for banning the gathering.
Golden Dawn, once a fringe movement, is now Greece's fifth-biggest party.
It was staging the handout to celebrate Greek Orthodox Easter, which falls this weekend. At past events, the anti-immigration party asked recipients to show ID cards proving they were Greeks.
Golden Dawn reportedly managed to hand out some food before police moved in, and it later continued the distribution elsewhere in the city.
Mr Kaminis said no party could use the square for such purposes and vowed to prevent "thuggery".
The mayor told the BBC that a Golden Dawn MP tried to attack him at a charity centre he was attending later in the day.
Bodyguards managed to restrain MP Giorgos Germenis, he said, although a 12-year-old girl was injured in the process.
Mr Kaminis says he plans to sue the MP over the incident, which was reportedly caught on camera.
Speaking to the BBC, the mayor called members of Golden Dawn "criminals".
The party, whose neo-Nazi ideology has been condemned by human rights groups, has soared in popularity during the financial crisis.
Greek authorities have been accused in the past of failing to take action against hate crime and attacks on immigrants.
Mr Kaminis said Thursday's police action was a "victory for the democratic state".
"Thuggery will not prevail in this city as long as I am mayor," he was quoted as saying by the Greek newspaper Kathimerini.
Police in the Greek capital Athens have stopped the far-right Golden Dawn party from handing out free food only to Greeks on the city's main square.