No, falsification is not required for a statement to describe objective reality (vs. objective morality). You want to look at Michael Polanyi's 1974 Personal Knowledge. On pages 43-48, he talks about crystallography, and how it is an unfalsifiable theory which is nonetheless empirically useful.
The principle of crystal symmetry was discovered by assuming that crystals contained only six elementary symmetries (mirroring, inversion, twofold, threefold, fourfold and sixfold rotations). From this it was concluded that the 32 possible combinations of these six elementary symmetries represented all distinct kinds of crystal symmetry.
The only sharp distinction laid down by this theory is that between the 32 classes of symmetry. They are distinct forms of a certain kind of order.
We may now turn to the question, on what principles our acceptance of crystallographic theory rests.
A classification is significant if it tells us a great deal about an object once this is identified as belonging to one of its classes. Such a system may be said to classify objects according to their distinctive nature. [...] Yet this system was supremely vindicated, as was the geometrical theory of crystals in general, by its classificatory functions. [...]
Here stands revealed a system of knowledge of immense value for the understanding of experience, to which the conception of falsifiability seems altogether inapplicable. Facts which are not described by the theory create no difficulty for the theory, for it regards them as irrelevant to itself. Such a theory functions as a comprehensive idiom which consolidates that experience to which it is apposite and leaves unheeded whatever is not comprehended by it. (44-47)
Contrast this to the following from Karl Popper's 1934 The Logic of Scientific Discovery. Karl Popper developed falsificationism.
In this formulation we see that natural laws might be compared to 'proscriptions' or 'prohibitions'. They do not assert that something exists or is the case; they deny it. They insist on teh non-existence of certain things or states of affairs, proscribing or prohibiting, as it were, these things or states of affairs: they rule them out. And it is precisely because they do this that they are falsifiable. If we accept as true one singular statement which, as it were, infringes the prohibition by asserting the existence of a thing (or the occurrence of an event) ruled out by the law, then the law is refuted. (48)
Popper talks a bit about axiomatic systems (e.g. p53), but for now I'll stop short of providing full Popperian reasoning for the folks who would give a 'Yes' answer.