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According to Newton, the force between two objects of masses $m$ and $M$ is $$ F = \frac{GmM}{r^2}, $$ where $r$ is the distance between the two objects. Basically, if distance between the object doubles, the force will reduce to the fourth of it. Also, the area of a 3-dimensional sphere is $$ A = 4\pi r^2. $$

My guess is that both the object "radiate" gravity particles (gravitons?) uniformly in all possible directions. Now, when these gravity particles have advanced from distance $r$ to distance $2r$, the result area of the "wave" will increase $2^2 = 4$ times, which will make that "wave" $4$ times weaker per area unit, which is reflected in the law.

My question is: can somebody tell whether my intuition makes any sense?

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Yes, you have essentially discovered Gauss's law for gravity.

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