MINNEAPOLIS—Chief Justice John Roberts on Tuesday sought to quell some of the furor surrounding the Supreme Court and the confirmation of Justice Brett Kavanaugh, telling a college audience that the court remains an independent, nonpolitical institution in which justices are committed to working together.

The chief justice, making his first public remarks on the Kavanaugh saga, took the stage in front of a packed auditorium of nearly 3,000 at the University of Minnesota and said he wanted to first address “the contentious...