The term appears in the International New York Times: "Radical Pope, Traditional Values":
As a result of its work in basic health and education — and despite its obtuse views on birth control — in the last 50 years the church has probably lifted more people out of poverty than any other civic institution in history.
Internet (I forgot where) suggests that a 'civic institution' by definition cannot be a government (or a market participant). However, we also have
civic Of, relating to, or belonging to a city, a citizen, or citizenship; municipal or civil.
civil (not comparable) Having to do with people and government office as opposed to the military or religion.
This could be construed as excluding the Catholic Church (and including government).
So, what, if any, is a good definition of 'civic institution'? Or is the term ambiguous?