Yes. That is the most basic application of Einstein's theories (among others...Einstein didn't do all the work!).
Imagine a large trampoline. Now put a bowling ball on the trampoline. The surface of the trampoline will curve downward under the bowling ball, and the surface of the trampoline will bend to a lesser extent in the area surronding it. Now put a marble on the trampoline near the bowling ball. The curvature of the surface of the trampoline will cause the marble to accelerate towards the bowling ball. If you couldn't see the trampoline, you might think that the bowling ball was somehow attracting or pulling on the marble (which is how Newton conceived of the motion in his Laws). But, once you can see the trampoline, you will realize that the marble is simply following the curvature of the trampoline. That is Einstein's great realization.
Likewise, if you were to roll the marble past the bowling ball, the path of the marble would be deflected by the indention in the trampoline's surface. You could view this as some force pulling on the marble, if you couldn't see the trampoline. But, upon seeing the trampoline, you would realize that the marble was simply following a straight line... it's the curved surface that caused the path to change!
The motion of the apple is "downhill" in the sense that its path is curved by space-time towards the Earth. We perceive that "path change" as falling. This is basic (well, intermediate) physics.
Then you posted . . .
"Gravity is due to space-time curvature."
(Seems to me you can't have 'em both)
Is it possible that energy is just simply attracted to energy?