The groups argued that regulators did not quickly process applications and had not contacted prospective voters who submitted incomplete forms.

The organizations alleged that the decisions essentially amounted to a lockout from the polls for thousands of Georgians.

But Judge Brasher wrote: “Not only is there no proof that the respondents have failed to fulfill their duties, but there is also affirmative proof to the contrary. The secretary of state and the county registrars have supplied evidence in their responsive pleadings, in their motions to dismiss and at the hearing on this matter that they have fulfilled, and that they are continuing to fulfill, their mandatory statutory duties regarding newly registered voters.”

The groups that brought the lawsuit immediately criticized the decision.

“It is a sad day when a court shows such a lack of concern for the disenfranchisement of citizens who are eligible and eager to vote,” Barbara R. Arnwine, the president and executive director of the Lawyers’ Committee for Civil Rights Under Law, said in a statement.