The word “dogma” that Feinstein deployed is specifically connected to the Protestant critique of Catholicism, and to its particularly nasty American version. A dogma is an article of faith laid down by an authority. One of the classic Protestant polemical attacks on Catholicism was the allegation that Catholics are obligated to believe what the church teaches them is incontrovertibly true, whereas Protestants are called on to form their own beliefs on the basis of individual faith and judgment.
Starting in the 1840s when Irish Catholic immigration to the U.S. increased sharply as a result of the Great Famine, American Protestants began to transpose this religious dispute into the realm of politics. Nativists argued that Catholics couldn’t be good citizens of a republic because they wouldn’t be able to form individual judgments about politics.
The shorthand version of this view asserted bluntly that Catholicism and democracy were incompatible — an argument that prefigures the contemporary view that Islam and democracy cannot coexist.