The court approved the participation of Jewish Power’s second-ranked candidate Itamar Ben Gvir as well as the Arab nationalist party Balad.
It is the first time it has used powers to prevent an individual, rather than a party, contesting the election since a change in the law.
Mr Ben-Ari condemned the court as a “judicial junta which seeks to take over our lives”.
Naftali Bennett, who left the Jewish Home Party to set up the New Right, also criticised the judges, saying they had crossed a “red line”.
But Israeli Labour leader Avi Gabbay welcomed the ruling, saying “good riddance” to Mr Ben-Ari.
Jewish Power’s involvement has led to arguments within the Board of Deputies with one group calling on the organisation’s leaders to condemn it but another telling the Board to remain neutral during Israeli elections.
