WASHINGTON—Supreme Court nominee Brett Kavanaugh wrapped up what is likely to be the toughest part of his confirmation process Thursday, completing two 13-hour days of public questioning in which senators often spent their time sparring with one another.

Senators battled each other over confidential documents, with Democrats threatening civil disobedience and Republicans warning of severe consequences, and Judge Kavanaugh for a time becoming a spectator at his own confirmation. That followed a Wednesday session in which Democrats...