No. Gravity is not the curvature of space-time. Space-Time curvature is the result of the existence of gravity. Actually, space is not literally curve. Only the moving object that creates a curve path. Or such moving object is expected to move in curve path. This is all due to gravitational pull toward the object where it revolves. That revolving object is pulled toward the inner core of an object where it revolves. The foot prints of a moving object creates the curve path . Or, the expected foot prints of a moving object is a curve path. The empty space surrounding around an object is not actually empty. It contains some so-called invisible matter or dark matter. The invisible matter can be thought like a pool of water or ether . Particles of dark matter serves as water or ether that makes all object in the universe suspended. So imagine a ball floating in the water not above water. Now, put a magnet inside the ball so that it will pull the object revolving around it. You can see the ball is pulling the object as it revolves toward its surface. That is, the magnet inside keeps on pulling it toward the ball's surface. Assuming the magnet is not strong enough such that the revolving object does not touch the surface of a ball; so, you will see the revolving object is somewhat flying above the surface of a ball. And this revolving object follows a curve path of a ball.
So, it is clear that space is not curve. It is the gravity that makes the path of a moving object to be curved. Now, if you insist, that space is curve. Actually, you may view the space is curve because the invisible suspending particles of dark matter are pulled toward the center of suspending object. This creates a curve like vision.
Or, all space points surrounding this suspending object are wrapping this object spherically. So at any point of this wrapping space, a moving object in this space, can be seen moving in a curve path. Take note this wrapping space is because of gravitational pull toward the inner core of such suspending object.
Now what makes this magnet inside or gravitational pull to exist?
Very simple example is in the small-scale objects. Like, the attraction or counter-attraction between proton and electron. Or, attraction or counter-attraction of particles inside sub-particles of sub-particles. It is normal for every particle to find stability with the other particle. It is instinct for each particle to find counter-balance so it becomes at rest; otherwise, this will continue to find its counter-balance until it gets one. If it could not find one, then, this will continue to move so it will become stable in constant motion mode. Example to that is the revolving planets around the sun, revolving moon around Earth. These celestial bodies are mutually and continuously finding their balance but until now they've not finished to get balance with each other. Hence, you can see the moon , the planets are continuously revolving.
What is this thingy that makes these two objects to have attraction or counter-attraction?
Example: Two atoms attracted to each other. What caused them to attract? Surely, one of the two atoms is unstable or loses its balance in its zone maybe because one electron transfer to other atom. This atom which loses its electron becomes unstable and wanting to get one electron so it will restore its balance or stability. Now, some atom is unstable because of extra electrons in their zone, this will make also the atom to continue to find its stability. The extra electron in their orbit disturbs the stability of such atom. Hence, it will continue to find its stability until it get one. So, once this unstable atom passes near other atom unstable atom, if one is lacking electron and the other has extra, they will attract. If one atom has extra electrons and meet with other atom of having extra electrons, they will repel. Or when both atoms have lack of electrons (both unstable atoms due to lack of electrons) then they still repel to each other. So the cause of attraction or counter-attraction are their being lack of or have extra of something that make them at rest or stable.
Now, how it could able to pull some thing to make its zone stable? Or , how it could able to push when the two unstable atom are lack of or have extra of something? As I said earlier, the pool of invisible particles in the dark matter serves as sliding plank toward the other. When this atom is unstable, the surrounding space or the wrapping space around that atom, makes this invisible water like stream or falls of dark particles to bring this electron toward the unstable of atom due to lack of electrons .The imaginary hole in the unstable atom is where the stream of water goes into. That stream carries the electron to shoot into that hole. So, it is the invisible water like pool in the empty space containing dark matter that makes the particle of white matter to swim or come along with it. (white matter is just used here to imply opposite of dark matter)