Ultimately, gravity has to obey wavelike behavior, even if it's not observable. Here is why. In order for gravity to obey the Einstein equations, every particle and quantum of mass-energy must emit a 4D spherical wave pattern that announces the particle's mass-energy content. These wave-patterns travel to the ends of the universe carrying gravitational potential energy information. And of course, as these wave-patterns expand, there gravitational potential becomes weaker by,
[math]U=-\frac{GMm}{r}[/math]
On the receiving end, every particle, wave and object receives a spherical gravitational potential wave-front that collapses down to the particle/object in question. This collapsing wave-front is carrying gravitational potential energy. Remember, these wave-fronts are traveling at the speed of light, and they are being emitted and absorbed continusouly. They are like a constant flow of gravitational information.
Call them wave-functions, gravity waves or whatever you like. This is how graviational information is transmitted and received. But this gravity flow is real. It's called the curvature of space-time and it explains how everything is connected to this universe.