If MOND were true, what would that imply for the geometrical description of gravity as curved spacetime? As I understand it, Modified Newtonian Dynamics, or MOND (Milgrom M., 1983, ApJ, 270, 365), slightly alters Newton's Law of Gravity by introducing a low acceleration limit below which (for an object in a circular orbit) the velocity no longer depends on the distance from the CM but rather only on the total mass of the system. As we know, Newtonian Gravity is the low-speed, weak-field limit of General Relativity. If MOND were found to be true, then it seems General Relativity would be inaccurate for certain systems. If MOND, not General Relativity, is the (more) accurate theory of gravity, what does that imply for the geometrical description of gravity as curved spacetime? Is that still valid?   
For example, scientists have detected gravitational waves that were predicted by General Relativity. I assume that if MOND does not incorporate curved spacetime, then gravitational waves would technically conflict with it. However, this doesn't seem likely, since the discovery of gravitational waves has been widely confirmed and accepted, and MOND is still being researched.  
What am I missing?
 A: General relativity is not the only possible theory of gravity, it is just the simplest such theory - or at least the simplest such theory that works. For many decades people have suggested modifications to GR by tweaking it or adding extra features and there is a long list of such theories in the Wikipedia article Alternatives to general relativity.
The original MOND theory from Milgrom is not generally covariant, but it has been made so by incorporating it into a theory called TeVeS. This is an extension of general relativity in the sense that it includes the action from GR but adds extra terms so it. TeVeS predicts much the same as GR in the scenarios we can test, as indeed it must do to be a viable theory, but it also matches the predictions of MOND.
So it isn't the case that if MOND were true it would invalidate general relativity, rather that GR would become an approximation that worked most of the time but would need the extra complexity of TeVeS outside the regimes where GR was a good approximation.
For the record we should note that dozens (literally) of alternatives to GR have come and gone but so far GR has survived every attempt to challenge it.
