Indeed, the modern Roman Catholic church should be protested if and when it attempts to interfere with the political process, or worm it's way in to government too much. Although we in the US don't have our own equivalent of Pussy Riot, we do have plenty of vocal citizens who speak out against the church when necessary (most recent example being when it tries to influence health care legislation, and coverage of birth control and abortion).
Same for the fundamentalist Protestant groups (religious right).
Luckily for all of us, the catholic Church seems to have (mostly) learned from history about what happens when the church and state are too closely intertwined,and these days usually limits itself to lobbying efforts. The church as a rule does not endorse candidates, but will criticize policy proposals (birth control coverage, the Ryan budget, for examples). The Russian Orthodox church, on the other hand, sounds like it is much more "in bed" with Putin and his government than the Catholic Church is anywhere, these days. plenty there to protest and criticize.