A federal judge denied a motion to move a legal showdown between Uber and Waymo into private arbitration, according to a court document filed Thursday evening, a decision that sets the stage for a public, bare-knuckle trial between the two companies.

It was a setback for Uber, which has been accused of stealing valuable technology from Waymo, the self-driving-automobile unit spun out of Google. Uber had pushed for arbitration, usually a less expensive and faster process — and one that neither takes place in front of a jury nor becomes part of the public record.

At the heart of the suit is Anthony Levandowski, a star engineer and a veteran of self-driving technology who Waymo has claimed stole technology from Google before leaving to form Otto, his own autonomous-vehicle start-up. Waymo has argued that Uber conspired with Mr. Levandowski to use those stolen files in Uber’s self-driving-car designs after Uber purchased Mr. Levandowski’s start-up for $680 million. Uber has denied the accusation.