WASHINGTON—The historic antitrust case against Google will soon be in the hands of a judge who says he has no idea how he will rule.

It will be many months before U.S. District Judge Amit Mehta, who oversaw the nonjury trial in Washington, rules on the question at the heart of the Justice Department’s case: Did Alphabet-owned Google, which answers about 90% of all internet search queries worldwide, illegally cement its monopoly?