TL;DR: it is the same as gravity, unless you’re defining ‘empty space’ in a really odd way. If fact, I can’t think of any way to experimentally prove this is not the case.
I think the key to this question lies in the definition of ‘empty space pushing’
For starters let’s assume space is quantised into a series of infinitesimal points. This isn’t strictly true, but it’s easier to conceptualise. Let’s assume each point of space that doesn’t contain matter pushes away from itself with a force that diminishes with the square of distance. It’s pretty easy to show (if we assume the universe is big enough that we can ignore the boundaries) that the net force at any given point is 0: all the points of nothing negate each other.
Let’s introduce a speck of matter.
Suddenly there is a space in the universe that is not pushing away from itself, which leads to an imbalance where the net force near this speck of matter is directed towards the speck of matter. The magnitude of this ‘not pushing away’ force is exactly the same as the original force you removed, ie inversely proportional to the square of distance.
Wait. That looks suspiciously like gravity.
If we continue to add specks then we’ll see the specks, responding to the imbalance in pushing forces, clump together. This leads to a larger volume of ‘not pushing’ which leads to a larger imbalance.
The same is true even if you take an Einstinian view of gravity, just replace ‘empty space pushes away’ with ‘empty space deforms space near it upwards’ (a strange concept, perhaps ‘outwards’ is a better word, but hey).