Stephen Bainbridge makes a strong case that they do. On admitting mistakes:

Let's assume arguendo that the ordinary universal Magisterium does evolve over time. Is Posner (and O'Neill) correct to think that this undermines the Church's authority? When all is said and done, after all, this is Posner's key claim.

I think the answer is no. Noonan gives three reasons to think this is so: (1) "Do parents lose or gain authority with their children when they admit to a mistake in guiding them?" (2) The American judicial system gains authority by admitting its mistakes. Indeed, contrary to Posner's apparent view as expressed by his reference to Supreme Court precedents, Noonan acknowledges that the Supreme Court makes mistakes and that its doctrines develop, but nevertheless fairly asks where "is the prestige of a judicial system higher than in the United States?" (3) No one loses respect for science, which is constantly changing its mind.

Noonan concludes: "Admitting error, the Church would not fare worse than parents, judges, or scientists, except perhaps among those who have conceived of the magisterium as a perfect machine perfectly enunciating moral truth in all ways at all times in all places."

Bainbridge also goes after Posner for asserting that the Catholic church is a corporation. 

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