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.

Officials in the Minnesota attorney general’s office did not immediately respond to a request for comment on the ruling.

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. Mr. Castro has not appeared in court to answer to those charges, and he did not respond to attempts by The New York Times to contact him in recent months at the jail where he has been held.

At the federal courthouse in Brownsville, Texas, lawyers for the two states argued on Tuesday about whether Mr. Abbott had a legal obligation to act on Minnesota’s extradition request before Thursday.

A final decision on extradition would be left to a state judge if the governor signed off on Minnesota’s request.

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. The Department of Homeland Security initially claimed that Mr. Castro had fired a shot in self-defense after enduring a sustained assault that involved a broom and a shovel.

Days after the shooting, Mr. Sosa-Celis and his roommate, who are both from Venezuela, were charged in federal court with assaulting a law enforcement officer. Federal prosecutors dismissed the case in February after investigators determined, based on a video of the encounter, that Mr. Castro had likely lied about the events that led to the shooting.

Mr. Castro was taken into custody in late May in South Texas and has been held in a county jail in Brownsville, near the border with Mexico, for nearly three months.

Soon after Mr. Castro’s arrest, Gov. Tim Walz of Minnesota, a Democrat, sent an extradition request to Mr. Abbott, who has so far refused to grant it. Such requests are routine when a defendant is arrested in another state, and they are generally granted without conflict. Defendants can contest an extradition, though they generally have narrow grounds to do so.

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.

Their conversations touched on plans “about marrying her and buying a house in Mexico when he is released,” according to the lawsuit.

Mr. Castro does not have a lawyer listed in the Minnesota case.

Mr. Abbott’s lawyers said that Texas had been reviewing whether Mr. Castro met the legal criteria of a fugitive under extradition law since he left Minnesota at the direction of his employer before he was facing criminal charges. The Texas lawyers also accused Minnesota of acting with an “odd rush to judgment” in its effort to return Mr. Castro to the state to face charges.

Judge Rodriguez said Mr. Abbott was well within his authority to review Mr. Castro’s eligibility for extradition, and that he had found no evidence that the governor was acting in bad faith.

Lawyers for Mr. Walz described Texas’ argument about fugitive status as a “smokescreen,” and warned that basic tenets of extradition procedure were under threat.

“Governor Abbott’s position, if accepted, would mean that any time a person committed a crime in one state and then left the state because of a job transfer, because of military orders, or even because they were shoved in a van, they could never be extradited to face prosecution,” the Minnesota lawyers wrote.

During Tuesday’s hearing, Trevor Ezell, Mr. Abbott’s general counsel, said his office was waiting for the Texas secretary of state to determine whether Mr. Castro met the criteria of a fugitive under extradition law.

He provided no timeline for the process, which Mr. Ezell said was being carried out in good faith.

Kimberly Svendsen, the lawyer representing Minnesota, questioned the delay to act, noting that the two states routinely heed extradition requests from one another expeditiously. “We have never seen an extradition handled like this,” she said.

Ethan Lowens, a fellow at the University of Pennsylvania’s law school who studies extradition law, called the dispute “extraordinarily unusual.”

In 1987, the Supreme Court established that the extradition clause of the Constitution empowers federal courts to force states to comply with extradition requests.

Mr. Lowens said that bitter political disputes over a range of issues, including immigration enforcement, abortion and transgender medical treatment, had set the stage for a new legal battle over extradition law.

“The politics of the moment makes it seem inevitable that a case like this is going to come up,” he said. “The law in this area is extremely underdeveloped and we have this political tinderbox of a moment right now.”