FBI agents arrested Nicoletti, Brian Carranza, 21, and Michael Contreras, 18, early Wednesday.
The three men pleaded not guilty in federal court to charges of conspiracy to interfere with voting rights.
"Violence and intimidation aimed at interfering with the constitutional rights of every citizen, including the right to vote, will not be tolerated," U.S. Attorney Benton Campbell said in a statement.
Nicoletti and Contreras were held without bail, while Carranza was released on $200,000 bond. Nicoletti and Carranza are white; Contreras is Hispanic. Their lawyers left court without speaking to reporters.
A fourth defendant was expected to separately plead guilty to unspecified charges, according to law enforcement officials, who spoke on condition of anonymity because the plea deal had not been made public.
Before the rampage on Nov. 4, 2008, the men had gathered at a makeshift clubhouse in their Staten Island neighborhood, where they monitored the election results on the Internet.
Prosecutors said in court papers that shortly after learning of Obama's victory, making him the nation's first black president, they "decided to find African-Americans to assault in retaliation for an African-American winning the election."
Nicoletti drove the group to a black neighborhood, where they stopped a teenage immigrant from Liberia, Ali Kamara, and beat him with a metal pipe and retractable police baton, prosecutors said.
They later pushed a black man to the ground, demanded a Hispanic man tell them how he voted and "yelled profanities about Obama as they drove past an election night gathering of African-Americans at a hair salon," the court papers said.
Finally, the defendants used their car to run over a white man, Ronald Forte, they mistakenly believed was black, the papers said.
The victim was in a coma for several days but survived.
Prosecutors also allege that last month, Nicoletti, believing Contreras was cooperating with authorities, went to his home with three friends and attacked him. Nicoletti punched Contreras and called him a "snitch," court papers said.
At the arraignment, Forte's mother told the court that her son had been "left in the gutter to die" and now requires full-time care.
"It's the most horrible thing that's ever happened to me or my family, and it's not going to go away," Eileen Forte said.
Kamara's mother also spoke, saying, "My son was innocent. They beat him for Obama."
Local prosecutors had previously charged two of the defendants with assault as a hate crime and weapon possession while the FBI investigated the civil rights case.
If convicted of the federal charges, the men each would face up to 10 years in prison.