Gravity as a force I want to know if my reasoning is correct.
If you drop an object at a height, the object accelerates downward at a constant rate due to gravity. Because we defined force as the ability to accelerate an object, gravity must also be a force.
And what if the acceleration was $0 \space m/s^2$, would gravity not be a force then, if my reasoning is correct? 
 A: You are asking what would happen if you put (zero relative velocity) an object (with rest mass) into the gravitational field of the Earth.
Now why does it start moving towards Earth? It is because the Universe is set up so and the four vector is set up so, that its magnitude for the massive object must always be c.


In physics, in particular in special relativity and general relativity, a four-velocity is a four-vector in four-dimensional spacetime[nb 1] that represents the relativistic counterpart of velocity, which is a three-dimensional vector in space.
    in short, the magnitude of the four-velocity for any object is always a fixed constant:
    {\displaystyle \|\mathbf {U} \|^{2}=c^{2}\,}  \| \mathbf{U} \|^2 = c^2 \,


https://en.wikipedia.org/wiki/Four-velocity
Now when you put the object into the gravitational field of the Earth, the gravitational field has an effect on the object, as per GR time dilation, the object will feel the passage of time slower relatively.
Since the object start moving in the time dimension slower (because it is in the gravitational zone of the Earth), but its four vector's magnitude needs to stay c, its spatial speed needs to compensate. It will start moving in space.
It will move towards the center of gravity.
As it moves closer to Earth, it moves deeper into the gravitational field. It feels even more gravitational potential (stress-energy).
Can it maintain a constant speed? No. It will feel even more gravitational potential, thus its speed will change, it will increase (accelerate).
Now you are asking can this object maintain a constant speed? The answer is no.
It is very important to understand that this is the reason why the object cannot keep a constant speed as it moves closer to Earth.
Speed is symmetrically relative. Acceleration is absolute.
As per the equivalence principle, the effect of the gravitational zone is the same as acceleration in terms of time dilation.


In the theory of general relativity, the equivalence principle is the equivalence of gravitational and inertial mass, and Albert Einstein's observation that the gravitational "force" as experienced locally while standing on a massive body (such as the Earth) is the same as the pseudo-force experienced by an observer in a non-inertial (accelerated) frame of reference.


https://en.wikipedia.org/wiki/Equivalence_principle
This is why we call gravity equivalent to the effect of acceleration.
Now you are asking for a gravitational zone, where if you let go of a massive object (zero relative velocity), the object will move toward the center of mass with constant velocity.
There is no such gravitational field in this universe.
A: Your statement

If you drop an object at a height, the object accelerates downward…

doesn't correspond to reality. If you are in a car and break or accelerate the car, you feel accelerations. If you fall from the sky, you don’t feel any acceleration, you are weightless (as long as the air drag doesn’t impeded the free fall). The same happens, if you are inside the ISS. From the earth I see you moving in a circle every 90 min around the world and comparing this with a weight on a rope, moved in a circle, you has to feel a huge acceleration. But you feel weightless. No force works on you.

Because we defined force as the ability to accelerate an object, gravity must also be a force.

Gravitation is not a force, it is the ability to bend the geodesic path of any object passing by. The geodesic path for objects with mass small compared to the celestial body depends only from its velocity. Photons have the least curved path with their unsurpassed speed of light. For heavy masses, such as the moon around the earth, the moon makes the earth a little wobbly, but measurable. But anyway, in all these cases the objects are not accelerated.
At the end its a simplification to talk about gravitational force. But at the end you are calculating something like instant velocity for some distance from the dropping or the angular velocity for the ISS or the revolution time around the sun.


Gravity is most accurately described by the general theory of relativity (proposed by Albert Einstein in 1915) which describes gravity not as a force, but as a consequence of the curvature of spacetime caused by the uneven distribution of mass. (Quote from Wikipedia)


