WASHINGTON—The Senate Judiciary Committee said Monday it would hold a hearing next week with Judge Brett Kavanaugh and the woman who has accused him of sexual assault, setting the stage for a dramatic public showdown that could determine the fate of his nomination to the Supreme Court.

The hearing will give citizens and lawmakers a chance to hear more about the three-decades-old accusations against Judge Kavanaugh from his teenage years at a Washington-area prep school. Judge Kavanaugh has repeatedly denied the accusation,...