For them, some of the major issues behind Pope Francis’ world-wide acclaim—his conciliatory approach to gay people, for instance—have instead been a cause for dismay. Like Ms. Dupuy, many fear the pope is blurring the lines around seminal teachings and creating confusion about what it means to be Catholic.

“Conservatives worry about the way he seems to have turned from the culture war over issues like abortion and homosexual marriage,” says Robert Royal, president of the conservative Faith and Reason Institute in Washington, D.C. “The image that gets transmitted is that these are not ‘Francis issues’—that he’s more interested in income inequality, the gap between rich and poor, the environment,” adds Mr. Royal, who belongs to the Catholic diocese here.

“It’s high time that he said, ‘here’s the church’s teaching and we will not change on these issues,’ ” says Patrick O’Neill, a father of three who attends another church in the diocese, Holy Trinity in Gainesville, and says he “humbly disagree(s)” with the pope on his decision to discuss allowing divorced and remarried Catholics to receive Communion.