Approval of Pope Francis among Roman Catholics in the U.S. has plummeted to an all-time low, according to a new survey, as the pontiff contends with a series of sex-abuse scandals in the church.

The survey released Tuesday by the Pew Research Center showed 72% of American Catholics now hold a favorable opinion of Pope Francis. It is the lowest approval rating of his pontificate, which began in 2013, down 12 points from the start of the year. His unfavorable rating, meanwhile, has more than doubled since January, to 20% from...