Is spacetime all that exists? Someone claiming to have studied physics is telling me that spacetime is all that exists and that this fact is the basis of modern physics. When I said matter/energy also exists in its various forms, he said that energy is merely spacetime "waving/vibrating/acting", else what did I think was doing the waving: magical fairy dust? and has presented the following Einstein quote to support his claim:
"When forced to summarize the general theory of relativity in one sentence: Time and space and gravitation have no separate existence from matter."
Does Einstein's quote support what he's saying, that space-time is all that exists, and that matter/energy is spacetime waving?
 A: 
When forced to summarize the general theory of relativity in one sentence: Time and space and gravitation have no separate existence from matter.

This reads to me as a statement of Mach's principle, probably from very early in the history of GR. Einstein originally thought that GR would embody Mach's principle. He was wrong, but it took decades for this to become completely clarified. For a longer discussion of Mach's principle, see this question: Is Mach's Principle Wrong?
As a counterexample to Einstein's claim, gravitational waves carry energy, and they exist independently of matter. (They don't even need to be generated by matter -- it's very reasonable to expect the existence of gravitational waves as a result of the Big Bang, and in fact in a maximum-entropy Big Bang, almost all the energy and entropy would have been in that form.) For precisely this reason, Einstein didn't want to believe that gravitational waves were real or detectable. He published an erroneous paper claiming that they were just a coordinate effect, and he never admitted later that the paper was wrong.
Even very early on in the history of GR, Einstein was dismayed by the existence of the Schwarzschild solution as an exact solution of the field equations. He didn't like it on Machian grounds, since it described a gravitational field surrounding a single pointlike object, but based on Mach's principle such a thing should have no physical meaning unless there were two objects interacting.

When I said matter/energy also exists in its various forms, he said that energy is merely spacetime "waving/vibrating/acting",

This is conceivable, but is considered unlikely. See the WP article on geons. As a simple counterargument, if spacetime is "waving," then we're talking about a gravitational wave. But gravitational waves carry no electric charge.
A: I don't think Einstein is saying anything about space time dynamics containing the description of matter and their properties. Or even about Gravitational waves existing independently without any matter being present. 
I think he is saying something very different all together( and I could be wrong). 
Space time is defined through how events co-ordinate, and an event by its definition implies that it is a material interaction. So any perception of space time and its properties implicitly contains within the limitation that we have to use matter and its properties to probe it. Thus spacetime is meaningless if it is seen to exist separately from the material whose behavior it describes.
