Prime Minister Jean-Marc Ayrault has asked the interior minister to take steps "immediately" to dissolve the Revolutionary Nationalist Youth (JNR).
Five people are under investigation over the death of Clement Meric, 18.
He was badly beaten in a clash between far-right and anti-fascist activists in Paris on Wednesday, and later died.
The Paris Prosecutor, Francois Molins said according to witnesses the two groups had run into each other by chance in a busy shopping district near St Lazare train station, where a fight broke out.
He said the arrested suspects claimed to have responded to provocation by the left-wing group.
All five suspects went before a judge on Saturday, two days after they were detained.
According to the prosecutor, the main suspect named Esteban, 20, was being investigated on suspicion of murder.
"The [suspect] named Esteban acknowledged to police that he had struck Clement Meric twice - bare-fisted, he claimed - including the blow that caused him to fall to the ground," Mr Molins said.
Other witnesses said "Esteban" had worn a knuckle-duster.
"A friend of Clement Meric said he saw him with a knuckle-duster, while another witness at the scene referred to a 'shiny object' in his hands."
Two sets of knuckle-dusters had been found at his home, the prosecutor added.
The teenager was placed on life support but was declared brain dead and died on Thursday.
French President Francois Hollande said he condemned the attack "in the strongest terms".
France has seen increasing tension between left and right, following a fierce debate over the introduction of same-sex marriage.
The JNR is the militant wing of a far-right movement known as the Third Way.
It has denied any link to the brawl. Its leader, Serge Ayoub, has been questioned by police following the killing but is not a suspect.
But Mr Molins said on Saturday that all the suspects under investigation had ties to the Third Way. Only one, a young woman named Katia, was said to have admitted being a member, he added.
Prime Minister Ayrault said in a statement: "I have asked [Interior Minister Manuel Valls] immediately to begin the procedure... to proclaim the dissolution of the Revolutionary Nationalist Youths."
The government has vowed to crackdown on fascist groups.
Mr Valls said: "Sadly such movements are resurging... racists, anti-Semites and homophobes".
France's main far-right party, the National Front, described the killing of Mr Meric as "appalling".
The French government is to take steps to break up a far-right group allegedly linked to the death of a left-wing activist.