I would argue the famous Michelson-Morley experiment.
A bit of background first before getting into the experiment itself. It is quite easy to surmise, via Newton's laws of motion, that Earth should be in a vacuum, otherwise the constant drag of traveling through some medium must eventually cause us to crash into the Sun. Despite this Isaac Newton himself proposed the concept of a Luminiferous Aether which pervaded all of space and was the medium through which light propagated. Before this point there were various concepts of an "Aether" pervading space, but this, I believe, was the first truly scientific approach to the concept as a way of explaining physical phenomenon rather than a simple supposition of existence (such as the Greeks had done). The Luminiferous Aether was proposed as an almost magical concept to avoid numerous physical issues. I think Wikipedia describes it best.
The mechanical qualities of the aether had become more and more magical: it had to be a fluid in order to fill space, but one that was millions of times more rigid than steel in order to support the high frequencies of light waves. It also had to be massless and without viscosity, otherwise it would visibly affect the orbits of planets. Additionally it appeared it had to be completely transparent, non-dispersive, incompressible, and continuous at a very small scale.
The concept of the Luminiferous Aether was accepted on the authority of Newton and inability to explain the propogation of light otherwise.
Michelson-Morley Experiment (1887)
It wasn't until the Michelson-Morley Experiment that the Luminiferous Aether was seriously shot down. The goal of the Michelson-Morley Experiment was to find evidence of this Luminiferous Aether and the null result was very strong evidence that this Aether did not exist.
The experiment itself was set up to measure the speed of light through this Aether. The idea was that as the Earth moved through the Aether, it would cause a sort of "Aether wind" that would slow down the speed of light. Thus if one measured the speed of light in the direction of the wind and perpendicular to it, one should get different speeds. The Michelson-Morley experiment set up exactly this scenario and used increasingly accurate measurements to try and find this difference in speed. Ultimate no difference was found and the Aether was ruled out as a material which existed.
From that point onwards it was assumed that space was a vacuum, completely devoid of anything. Oddly enough, this assumption was so strong, that people didn't even really believe in the concept of the Solar Wind at first and were quite perplexed initially when they sent the first rockets into space with particle detectors that ended up detecting all sorts of charged particles in space. Regardless, I would say the Michelson-Morley experiment is the first time scientists had scientific evidence that space was a vacuum.