Going through the internet I've found claims (like the ones found on this MinutePhysics video) that Newton's law of gravitation could not be applied to small, everyday objects. However, after reading about the Cavendish Experiment for calculating the gravitational constant, how could these claims be considered true if at Cavendish's time the understanding of gravity was based solely on Newtonian Gravity?
P.s: If possible please mention scientific sources.
The video you link is making the point that we cannot measure the gravitational force if the masses are too small. Cavendish used two lead spheres weighing $158$kg as his large masses, and even though gravity is a weak force this mass is large enough to produce a detectable force. But suppose the masses were $158$g, $158$mg or $158\mu$g. At some point the gravitational force is going to get too small to be measured, and if we cannot measure the force we cannot know for certain that it obeys Newton's equation.
In fact there have been suggestions that the inverse square law might break down at very short distances. This would happen if there were large extra dimensions as suggested by some string theorists. Experiments have been done to try and measure the gravitational force at short distances and no deviation from Newtonian gravity has been seen.
Anyhow, the video is not saying that Newtonian gravity does not apply to very small objects, only that we cannot experimentally test whether it applies or not. However barring the aforementioned string theorists I know of no physicists who seriously believe Newton's law fails for small masses.
While not directly relevant I should mention that there have been suggestions that on astronomical scales Newton's law fails when gravity is extremely weak. This is the principle behind Modified Newtonian Dynamics (MOND). The jury is still out on this though the consensus appears to be that MOND is not correct.
