The "American Catholic Church" (or "AmChurch" as some call it) refers to the Catholic experience in America of practicing the Faith with a "don't tell me what to believe" type attitude that is characteristic of Protestantism. The term is typically used by Traditionalists when speaking of “Liberal Catholics” in America, but the problem is much more serious because the Tradosphere is actually founded upon essentially the same erroneous (Liberal) principles as the AmChurch’s 'left wing'.
Long ago Rome saw these dangers creeping in, and so Pope Leo XIII issued an Encyclical against "Americanism" in which he singled out three big dangers we face:
These dangers are: (1) the confounding of license with liberty; (2) the passion for discussing and pouring contempt upon any possible subject; (3) the assumed right to hold whatever opinions one pleases and to set them forth in print to the world. (Testem Benevolentiae Nostrae)
The first error is the chief characteristic of Liberalism, which states that man is not bound to any law beyond himself, including not bound to Natural Law. The result is that "Liberty" becomes defined as “the right to do or say whatever you want” - which is not true Liberty at all, since it lacks restraints.
Building on the first error, the second and third AmChurch errors listed above immediately find their justification. They now think they have the right to discuss, criticize, pour out contempt upon, and especially publish anything he pleases. This "assumed right" states that one can even smear the reputation of another, protected under the guise of "free speech" (another Liberal error).