WASHINGTON—The Supreme Court, in an anticlimax, sidestepped a sweeping ruling on affirmative action Monday, directing lower courts to re-examine whether a race-conscious admissions program at the University of Texas at Austin should survive constitutional scrutiny.

The 7-1 ruling reflected a broad consensus across the court's ideological spectrum, but the justices reached near unanimity only by bypassing the core question in affirmative-action cases—whether racial diversity qualifies as a compelling government objective justifying...