The Volokh Conspiracy
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Many courts, including in Ohio, do allow permanent injunctions against repeating material that has been found, at trial, to be libelous. But "before a court may enjoin the future publication of allegedly defamatory statements based on their content, there must first be a judicial determination that the subject statements were in fact defamatory," in the words of Bey v. Rasawehr (Ohio 2020). And courts throughout the country have confirmed that this means a finding at trial, not just at an abbreviated pretrial hearing. To quote a Kentucky Supreme Court decision, for instance,
[C]onsistent with the modern rule, we construe Section 8 [of the Kentucky Bill of Rights] as permitting an injunction against false, defamatory speech, but only upon a final judicial determination that the speech is false. "A party may obtain injunctive relief in the circuit court by permanent injunction in a final judgment." Until such determination of falsity, however, the provision is best interpreted as proscribing a preliminary restraint upon the alleged defamatory speech. We hold that neither a restraining order … nor a temporary injunction … may be used to enjoin allegedly defamatory speech.
The corresponding Ohio Bill of Rights provision is much like the Kentucky one, and more broadly this is understood as a federal constitutional principle as well.
You can read the motion for the temporary restraining order, but nothing in it deals with the constitutional question. The judge's order denying the motion didn't offer a detailed analysis, but just said (as is common for state trial court orders in such situations):
PURSUANT TO MOTION FILED. EX PARTE HEARING HELD ON 10/05/21. PLAINTIFF'S MOTION FOR EX PARTE TEMPORARY RESTRAINING ORDER IS DENIED.
HEARING SET FOR 10/13/2021 AT 01:00 PM.