I think space itself is some kind of medium and not necessarily matter based. Currently, its composition is unknown. You can call this medium aether if you like, just do not confuse it with the luminiferous aether. There are several aether theories out there, one of the popular ones is the Superfluid Vacuum Theory.
Now, back to gravity. We know since Einstein that matter curves/disturbs space**. So, this medium, which is disturbed by matter, wants to be 'smooth' and it exerts a force on matter. This force is obviously a push force and we call it gravity.
From our perspective there is no way to differentiate between pull or push. Maybe until now.
** I know that according to GR gravity is a curvature of space-time and not just space. But if space itself is some kind of medium then the time dimension simply does not exist. What happens is that the disturbance of space prevents matter particles to interact with each other. We perceive this lack of interaction as if 'time' slow down since there is no change (We can only perceive 'time' by observing change). This picture is compatible with GR since relativity between moving bodies through space remains. However, it establishes a distinguished reference frame which is space itself.