"Politically motivated violence of this kind is unacceptable anywhere, and history has shown the grim consequences if it goes unchecked," said Jezerca Tigani, Amnesty's deputy Europe and Central Asia program director. "The Greek authorities must send a clear message that attacks like this will not be tolerated."
The suspect, who was not named in accordance with Greek law, appeared before a prosecutor Wednesday evening along with his wife, who was arrested on suspicion of concealing evidence. Another couple also appeared in court on similar charges. Five prosecutors have been assigned to the case.
Golden Dawn, whose senior members have expressed admiration for Adolf Hitler although they deny being neo-Nazi, won nearly 7 percent of the vote in 2012 general elections. Recent opinion polls show its support has since risen to around 12 percent.
Party members and supporters, often clearly identifiable in black T-shirts and combat pants, have been suspected of beatings and stabbings across the country, usually of dark-skinned migrants. In January, two men identified as party sympathizers were arrested for the fatal stabbing of a Pakistani migrant worker.
But Wednesday's killing was the first attributed to a Golden Dawn member, and the most severe attributed to political rather than racial motives.
"I am shaken by the event," said Public Order Minister Nikos Dendias, who cancelled a visit to Rome scheduled for Thursday. The killing and other recent violent incidents "show in the clearest possible way the intentions of the neo-Nazi creation."
Michaloliakos, the Golden Dawn head, said his party "unreservedly condemns the murder of the 34-year-old at Keratsini and denies any involvement of the party."
"All the political parties must assume their responsibilities and not create a climate of civil war, giving a political character to a tragic event," he said.
Police spokesman Christos Parthenis said the suspect drove to the scene of an altercation between two groups of people, got out of the car and stabbed Fyssas. Friends of the victim told Greek media they had been attacked by a large group of men as they left the cafe.
Golden Dawn lawmaker Michalis Avranitis offered a different version of events, saying the victim and the suspect had initially argued about a soccer match.
"Yes, this man, as it turns out, has declared himself to be a member of Golden Dawn. But Golden Dawn has 1 million supporters," Avranitis said in Parliament. "If, in a restaurant, two drunken idiots have a fight and someone is stabbed, should we look at their ideology and blame that?"
Elena Becatoros reported from Athens. Costas Kantouris contributed from Thessaloniki.