Gov. Andrew M. Cuomo must soon schedule a special election to fill the vacant congressional seat representing Staten Island and parts of Brooklyn, or a federal court will do it for him, a judge ordered on Tuesday.

The House seat for the 11th Congressional District became vacant in January, when Representative Michael G. Grimm, a Republican, resigned after pleading guilty to tax evasion. It is up to the governor to set a date for an election to replace Mr. Grimm, but Mr. Cuomo has thus far given no indication of his intentions.

In his ruling, Judge Jack B. Weinstein of Federal District Court in Brooklyn wrote that Mr. Cuomo had offered “no justification for his failure to issue the writ of the special election, much less any hardship preventing him.” Absent a fresh rationale for the delay, the judge gave Mr. Cuomo, a Democrat, until noon on Friday to choose a date, or “this court will fix the date for a special election as promptly as the law will allow.”

Melissa DeRosa, a spokeswoman for Mr. Cuomo, said he would name a date for the election in due course.