I hate that curved space graphic. Now you have some mysterious gravity pulling both the sun and earth down. I wish someone would come up with a better example to show gravity as curved space.
But gravity isn't curved space, it's curved spacetime. Think of a graph that shows both space and time, so that it shows constant motion as a straight line (four units of space per one unit of time with the slope constant). Then acceleration is a curved line. Now curve the graph paper - not through some higher dimension, but by stretching it and compressing it in ways that change the shortest distances between two points - so that the Earth orbits the Sun in a straight line. Not a straight line in space, mind you, but a straight line in spacetime: no acceleration. After all, since no one is pushing on the Earth, it should move in a straight line, right? Got it?