While there's no currently known answer to your question (since we don't have a quantum theory of the true vacuum and of gravity), I'm tempted to speculate a bit and very roughly (for what it's worth):
We usually think of matter as something placed "inside" space, and to define vacuum as the absence of matter/energy. Maybe it's better to interpret matter as "vacuum removed".
You start with an empty universe. There's a flat geometry vacuum. Vacuum isn't the same as nil, since there is geometry (lengths, time duration, volume, etc). In a sense, vacuum is a bit like some kind of weird inert fluid, that has the same properties for any observer (the fluid doesn't "flow": it doesn't have a local velocity for any observer).
Then add a small ball of matter. This ball "punches a hole" in the vacuum, since now there's something siting in space: there is less vacuum where the ball is, and the amount of vacuum removed is defined by the ball's energy content. Creating the ball is like "pushing" the vacuum to the outside, i.e. produces some deformation (curvature!) in the vacuum geometry. We could argue that gravity is a macroscopic effect of removing some vacuum by adding energy in it.
Notice that empty space may have a residual energy density everywhere (the cosmological constant), even in the absence of any content. I could elaborate on this idea but I think this is out of the scope of your question...
This "answer" may feels like pure non-sense speculation (for what it's worth!). Well then, too bad for the down votes.