Marcus J Ball brought a private prosecution against Mr Johnson over his claim during the 2016 campaign that the UK sends £350 million a week to the European Union.
A summons was issued for the Tory leader and new PM to face allegations of misconduct in a public office before it was quashed by High Court judges.
Mr Ball said today that his team has filed for permission to appeal to the Supreme Court over the decision.
He said: "The High Court's decision is far-reaching. It should be of great concern and interest to the public as a whole.
"We have grave concerns that public officers can now abuse their status and influence and to act in blatant disregard of the duties and obligations placed upon them, safe in the knowledge that no criminal consequences will follow."
High Court judges last month ruled the decision to issue the summons by a district judge was incorrect.
Lady Justice Rafferty said the problem of false statements during political campaigning is not new and that laws were in place to deal with "certain false campaign statements which it considers an illegal practice".