Are social encyclicals binding? Not everything in an encyclical—social or otherwise—is equally binding. Catholic teaching itself distinguishes different levels of authoritativeness for different kinds of teaching and different kinds of Church pronouncements. Some teaching is de fide (of faith) and must be accepted with “the assent of faith.” Such teaching is binding in an absolute and irrevocable way. Below that is teaching which, while not de fide, is nevertheless authoritative. Such teaching must be accepted with an obsequium religiosum, usually explained to mean “a religious assent of intellect and will.” Authoritative teaching is also binding, but not in an absolute and irrevocable way. One can entertain as a real though remote possibility that the teaching is false, but the benefit of the doubt goes to the Church, and there is a strong presumption that the teaching is correct. Such authoritative-but-not-de-fide teaching is like that of good parents: it may not be infallible (as de fide teaching is), but it is highly reliable and one is subject to it.
Below that is teaching which, while not authoritative in the above senses, is nevertheless owed great respect. In an encyclical—any encyclical—one finds statements of various levels of authority. It is not a simple question of whether “an encyclical is binding”.
It is vaguely analogous to a decision of the Supreme Court. The holding of the Court is binding, but not everything written in the majority opinion(s) is equally authoritative.
In a social encyclical, one finds statements of general principles. These are the most authoritative. One also finds various analyses of particular political, economic, and social situations. These usually involve judgments of a prudential sort that are not binding in either the de fide or authoritative sense. They still merit respectful attention, as coming from the supreme earthly shepherd of the Church.
For example, the principles of subsidiarity and solidarity, the existence of and limits to the right to own private property, the principles regulating just warfare, and the like are highly authoritative. But the more one descends to particulars, the less one is dealing with “binding doctrine” and the more one is dealing with pastoral guidance.
I do think that it would be better if Catholics were not so disposed to pick these documents apart like an English teacher grading a student paper. A little more obsequium would be nice, even as we recognize that not everything in these documents is of equal weight.
If you want to read more about the different levels of authority, one can begin with Lumen Gentium (a document of Vatican II). There was also a document of the Congregation of the Doctrine of the Faith a few years ago called Ad Tuendam Fidem, which is very helpful. A good discussion of the traditional categories of teaching is given in the beginning of Ludwig Ott’s Fundamentals of Catholic Dogma. While written in the 1950s, his discussion is not essentially different from that of Vatican II and post-conciliar statements.