A federal judge on Wednesday sided with Texas over Minnesota in a rare legal battle over extradition, with Minnesota seeking the return of an immigration agent facing state assault charges related to an on-duty shooting last winter. The case pitted one of the country’s most prominent Democratic governors against a leading Republican, and it raised a question that federal courts have wrestled with only a handful of times before: How much discretion do governors have over extradition requests from another state?

Judge Fernando Rodriguez Jr. of the U.S. District Court for the Southern District of Texas sided with the Republican, Gov. Greg Abbott of Texas, finding that Minnesota had not proven that Mr. Abbott had acted inappropriately in waiting to act on the extradition request. The ruling by Judge Rodriguez, who was appointed to the bench by President Trump, means that the immigration agent could be released from a Texas jail on Thursday after three months in custody. Texas law limits detention on extradition warrants to 90 days.

The agent, Christian Castro, was charged in mid-May in state court in Minneapolis with four counts of assault and one count of falsely reporting a crime. State prosecutors in Minnesota accused Mr. Castro of shooting through the front door of a Minneapolis duplex on the evening of Jan. 14 after an immigrant he had been chasing ran away after a car chase. Mr. Castro was accused of shooting and wounding Julio C. Sosa-Celis, the roommate of the migrant who had fled.

Gov. Tim Walz of Minnesota, a Democrat, sent an extradition request to Mr. Abbott, who has so far refused to grant it. Last week, Minnesota filed a lawsuit in federal court seeking to compel Mr. Abbott to sign off on Mr. Castro’s extradition. Minnesota officials expressed concern that the agent could flee to Mexico if he were released, citing monitored conversations he has had while in custody with a woman south of the border.