LaRose says he will appeal the ruling issued in Cleveland
The Ohio requirement applies when voters register at a motor vehicle bureau. The rule took effect for the first time this year.
Oliver issued the temporary injunction in Cleveland. His ruling said the National Voter Registration Act takes precedence over the Ohio requirement and that the state rule appears to conflict with the federal law.
The order suspends enforcement of the documentation requirement ahead of the Oct. 5 deadline for registering for the fall midterm elections.
The lawsuit was brought on behalf of the Ohio Alliance of Retired Americans and Red Wine & Blue by the Elias Law Group. LaRose, a Republican, said he would immediately appeal.
The ruling comes as President Donald Trump pushes Congress to pass a bill that would require documentary proof of citizenship to register to vote, along with other election changes.