Is the gravitational force a phenomenon or a theoretical invention? We use to say that gravitational force is a phenomenon. But since the theory of General Relativity replaced Newtonian gravitational force, how is it possible for a phenomenon to be replaced by a theory? Both are not phenomena we observe. What we observe is the free-fall, the movement of Moon around the Earth etc and we infere the theories of Newtonian gravitational force and the General Relativity. Is it correct?
 A: 
how is it possible for a phenomenon to be replaced by a theory?

Phenomena are described by theories. They may be a good description, highly confirmed by experiments and observations, like (as far as we know) General Relativity is for the phenomenon of Gravity. They may be a good description for a certain realm, where we make some limiting assumptions on what the theory can and can't predict, like Newtonian Gravity. And of course, some theories can be outright wrong, having no relation to experimental observations whatsoever.
Also, since the question is about gravity, it would only be fitting to quote Einstein:

The theoretical scientific researcher is not to be envied, because Nature—or more precisely put: experiment—is a merciless and not very kindly judge of his efforts. She never says “yes” to a theory, in the best case merely “perhaps” but in most cases simply “no”.

So it should be added, not only that theories don't replace phenomena, but we can never hope to prove that any theory is even a completely correct description for any phenomenon.
A: Short answer
Newtonian gravity and general relativity both agree that gravity will push objects around, as we experience in everyday life. But they differ in how they describe this pushing-around:

*

*In Newtonian gravity, it is described as a force, given by $F = G\frac{m_1m_2}{r^2}$.

*In general relativity, it is instead described as a "fictitious force" that has something to do with "the geometry of spacetime" (whatever that means). That is why you hear claims like "gravity is not a force in general relativity".

Why fictitious force?
To see why one might describe gravity as a "fictitious force" instead of an actual force, let's consider another gravitational phenomenon, frame dragging:

*

*We know that time runs slower close to a large mass (e.g. a black hole); this time dilation effect is obviously not a "force" in the Newtonian sense.

*Now suppose an observer Alice is moving parallel to a wall of mass at speed $v$. At some point, she drifts a bit closer to the wall. As she gets closer, gravity gets stronger, so her time runs even slower. Thus she slows down relative to the wall, say to speed $0.9v$. ((Speed = distance/time) decreases as time increases.)

*Finally, consider the perspective of a distant observer Bob who is also moving at speed $v$. From Bob's perspective, Alice is standing still while the wall is moving at speed $-v$. Then when Alice drifts towards the wall, she "slows down" to speed $0.9v - v = -0.1v$: she is "dragged" in the same direction as the wall.

To Bob, it looks like Alice has experienced a force in the $-v$ direction. But from steps 1 & 2, we know that this is really just a consequence of time dilation, which isn't a force. Hence it is natural to call this force "fictitious".$\mbox{}^1$
General relativity takes this further and describes all gravitional forces as fictitious. Indeed, the math doesn't really distinguish between effects like frame dragging vs ordinary pushing-objects-around - they are both consequences of a single rule, "objects follow geodesics" (whatever that means).
Alternatives
General relativity's choice to describe gravity in terms of fictitious forces is really a presentation choice, not a physical law. For example, teleparallel gravity (which is equivalent to general relativity) does describe pushing-objects-around as an ordinary force, separate from fictitious forces like frame dragging.
$\mbox{}^1$Technically, "fictitious force" has a more precise meaning than just "a non-force that pushes things", but that captures the spirit.
A: Gravity is a phenomenon we observe in this universe , but the "theoretical inventions" are the explanations given to gravity .Both Newton's laws of gravity and general relativity explain the same gravity but in a different way(the former as a force and latter as a distortion in space)!
A: 
how is it possible for a phenomenon to be replaced by a theory?

Theories don’t replace phenomena, they explain them. Newton’s law of gravitation and general relativity both explain the same phenomena. The explanations differ, but in many cases they are equivalent. In some circumstances they are not equivalent, and in those circumstances GR is more accurate.
A: Gravity is an interesting phenomenon that still needs plenty of investigating. Relativity helps calculate or predict what gravity will do but does not explain how or why matter can bend space time. We still need a theory to physically explain what's happening or how gravity works.
A: Let's take a characteristic example, Newton's gravitational law.
Newton found a mathematical relationship between two (observable) phenomena: the free fall of an "apple" on the Earth's surface and the movement of the Moon around the Earth. He found that for both phenomena the products between the acceleration and the square of the distance from the center of the Earth are the same. So we don't have anything more than some phenomena and a mathematical relationship between them.
Is this trivial? Not at all, because Newton claimed, and has not been disproved, that this is a universal law.
Is this an explanation of the phenomena? In a sense yes: He found that these apparent different phenomena and many others, such as planetary motion, have something in common.
What do all these phenomena have in common, apart from the aforementioned mathematical relationship? Newton gave the answer: "Hypotheses non fingo" or "I contrive no hypotheses".
So at the end of the day what we left with is an exact universal mathematical law between observable phenomena and some kind of helpful "explanation".
Then what is gravity? It is just a word which informs us that we deal with phenomena, where Newton's universal law holds.
