WASHINGTON — After a federal judge unsealed a search warrant this week that the F.B.I. had used to examine emails related to the previously closed inquiry into Hillary Clinton’s private server, some of her backers said it proved their contention that the warrant should never have been sought because it would not have changed the outcome of that investigation.

The decision by the bureau’s director, James B. Comey, to make the new email examination public in the presidential election’s closing days was a “stain” on his legacy and amounted to an “extraordinary impropriety,” Mrs. Clinton’s supporters said. Mrs. Clinton herself has said that Mr. Comey’s move cost her the election.

But while the F.B.I. ultimately did not find anything to alter its decision in July not to pursue criminal charges against Mrs. Clinton or her aides, agents believed they needed to examine the emails to be certain there was nothing new in them. This is common procedure for the bureau, whether the case involves a former secretary of state like Mrs. Clinton or anyone else.

In seeking the warrant, agents told the judge there was reason to believe the new emails might contain classified information. The emails were found on the laptop of Anthony D. Weiner, the estranged husband of Huma Abedin, a top Clinton aide. The laptop was not authorized to store classified information. The judge said there was sufficient probable cause and signed the warrant on Oct. 30.