When was it proven that the earth hung on nothing?

The earth isn't "hung on nothing". This passage from the book of Job implies some sort of need for the earth to be suspended from the heavens. But this is incorrect, the earth isn't hung at all.

The closest that we could say to the earth being "hung" is that it is in a system with the sun, where the two bodies interact with each other by the force of gravity. If this is what the passage was referring to, then it is incorrect, as the two bodies are tied by gravity, and not nothing.

But perhaps the writer of Job meant that the earth doesn't hang at all, and if this is what he meant, we should note that he had a 50/50 chance of getting this correct. And obviously this was the way he was going to go, since there is nothing visible that the earth hangs on, and other heavenly bodies, like the earth and sun, appear to hang on nothing also.

But Job tries to cover a few more bases, by contradicting himself. In Job 9:6 he says "He shakes the earth from its place and makes its pillars tremble". Now apologists claim that the numerous mentions of the earth being set on pillars are metaphorical, but the solitary one of it hanging from nothing is literal. Whether or not this is reasonable I will leave for the reader to decide.

As far a science goes, the ancient greeks thought the earth was spherical about 600 BC, presumably this means it is not hanging on anything. Aristotle later had a few arguments for it, and by the third century BC it was pretty much accepted. Modern scholarship places the book of Job as written around these times also (source: Eerdmans Bible Commentary), although christian literalists place it a thousand years earlier.

Newton first described gravity in the 17th century, and then Einstein's Theory of General Relativity explained how gravity can be explained by the curvature of space in the early 20th Century.

I hope that helps.