Newtonian gravity yields specific observable consequences, the most striking of which is the emergence of a 1/r^2 force. In so far as communication can arise via such interactions between distant particles, we can ask what would be expected for a theory of gravity that only allows classical communication. Many heuristic suggestions for gravity-induced decoherence have this restriction implicitly or explicitly in their construction. Here we show that communication via a 1/r^2 force has a minimum noise induced in the system when the communication cannot convey quantum information, in a continuous time analogue to Bellʼs inequalities. Our derived noise bounds provide tight constraints from current experimental results on any theory of gravity that does not allow quantum communication.