I'll stay in the realm of the classical (Newtonian) theory of gravitation, as you specified. I therefore think that GR is off-topic in this thread.
Physicists have established that any two object possessing mass interact, that is influence each other's movement. From this discovery steams the concept of gravitation and gravitational interaction. The identification of this phenomenon does not require the existence of situations where such an interaction does not exist.
The gravitation does have "particulars". The "particulars" of gravitation are interactions of any pair of objects possessing different masses (or mass distributions) and/or situated at different relative positions. (The dependence of the gravitational interaction on only these two characteristics is established by observation.)
Physicist have introduced concepts and tool to characterize quantitatively this (and other kind of) interactions. The gravitational interactions are characterized quantitatively by essentially the same concepts and tools as other kinds of interactions, be they of the action-at-a-distance kind or contact interactions. You singled out the latter type (a person pushes a block, etc.) and labeled it as somehow more "real", while in fact there is no reason to do this.
The tools physicists have introduced in order to characterize interactions of any kind are forces – described by vectors and tensors; it is understood that we are talking here of macroscopic, non-relativistic and non-cosmological realm (I am being super-cautious ). Correspondingly, both contact and longer range interactions, including the gravitational one, are described in a unified manner. In all this cases "the force is caused by some object acting on another" and would not exist in the absence of that object. In the case of longer-range forces, they would decrease with the distance between the objects, but not always totally disappear.
Gravity is in some ways similar to the electrostatic force; I am not sure if you see the same conceptual difficulties also for the latter. Gravitation has some unique characteristics, but I will not go into this now because, as I said, I don't know if this is relevant for you - for your subject, that is.
PS: Gravitation is perceptually detectable
PPS: Gravitation is caused by the mass characteristic of the objects. This is demonstrated by the fact that the less massive an object is, the less intense does it gravitationally interact with other objects.
PPPS : In GR gravity is caused by object's mass, but not directly as in the classical theory, but indirectly, by curving the space-time at the place where the second object exists.