WASHINGTON—The fight over Supreme Court nominee Brett Kavanaugh took another unexpected turn Friday when Sen. Jeff Flake forced a delay of up to one week on a Senate vote to allow the FBI time to investigate decades-old allegations of sexual assault against the judge.

In a tumultuous day, Mr. Flake (R., Ariz.) both provided the key vote to advance Judge Kavanaugh through the Judiciary Committee and used the threat of later withholding his support to force GOP leaders to slow the confirmation process down.

“This...