Supporters of Richard Spencer are suing public universities in Ohio, Michigan and Pennsylvania, seeking to force the schools to allow him to speak on campus – and to provide security to allow the far-right provocateur to “speak safely”.

The latest lawsuit, against Ohio State University, was filed on Sunday, days after police in Florida announced attempted homicide charges against three men who shouted “Hail Hitler!” and fired a gun at protesters after Spencer spoke at the University of Florida.

Security for the speech in Gainesville on Thursday, which drew more than 2,500 protesters, cost at least $600,000, the university’s president said.

Ohio State, Penn State and Michigan State have announced they will not allow Spencer to rent a campus venue, citing public safety concerns and serious disruption to school communities.

Spencer’s backers argue this is an unconstitutional infringement of his rights, since the government, including public universities, is not allowed to discriminate based on the content of speech. In April, Spencer won a suit against Auburn University in Alabama, when a federal judge ruled a planned speech should be allowed.

“While Mr Spencer’s beliefs and message are controversial, Auburn presented no evidence that Mr Spencer advocates violence,” US district judge W Keith Watkins ruled. “Auburn did not produce evidence that Mr Spencer’s speech is likely to incite or produce imminent lawless action.”

The lawsuits in Ohio, Pennsylvania and Michigan will test whether the attempted homicide charges in Gainesville, as well as a violent attack on protesters at a far-right unity summit in Charlottesville, Virginia, in August, which left more than 19 protesters injured and one dead, have made any difference.

Ohio State, which had delayed giving a final response to the request to rent a venue to speak on campus, wrote in a letter to an attorney for Spencer’s college tour organizer late on Friday night that they would not allow him to rent a venue, citing “yesterday’s events at the University of Florida”.

Gainesville police arrested three white men for attempted homicide. All had links to white supremacist groups and previous white supremacist events, according to the Anti-Defamation League’s Center on Extremism, and had given media interviews talking about their beliefs outside Spencer’s speech.

“I’m disappointed in the course of things,” Tyler Tenbrink, the alleged shooter, reportedly told the Gainesville Sun before the shooting. “It appears that the only answer left is violence, and nobody wants that.”

Less than two hours after Spencer’s speech, the men got into an altercation with a group of protesters after yelling “Hail Hitler!” at them and giving the Nazi salute. As two of the men yelled “Kill them!” and “Shoot them!”, police said, Tenbrink fired a single shot at the protesters. Police said he had a felony record that barred him from owning a gun.

The round hit a building. As the men drove away, a protester noted their license plate and called the police. All three were being held in the county jail.

Michael Carpenter, a lawyer representing Ohio State, wrote on Friday that “the university values freedom of speech” but that it was denying the request to let Spencer speak “due to substantial risk to public safety, as well as material and substantial disruption to the work and discipline of the university”.

Kyle Bristow, a Michigan attorney who is a white nationalist and “in-house attorney for the alt-right”, filed a lawsuit against Ohio State on Sunday.

On Twitter on Friday, Bristow wrote in Latin: “The die is cast.” In a press release on Sunday, he saud: “The alt-right shall enjoy the right to free speech.”