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I have heard many people say that the existence of atoms is proven by Brownian motion. Now, I understand how an atomic theory would suggest the existence of Brownian motion. However, who is to say that there is not another theory for what our world is composed of that can also predict Brownian motion (as well as the other phenomena predicted by atoms)? Of course, I am not sure what that theory would be, but I am wondering how one could say that Brownian motion proves the existence of atoms.

A quote from a physics textbook, which I read years ago, has always stuck in my head. It said:

"Some of the material in this book is undoubtedly incorrect. But if we are really lucky, MOST of it is incorrect".

This is because, in one sense, you are getting it spot on in your question.

There IS a theory that describes Brownian motion more accurately than our current model of atoms jostling a given particle.

This theory would improve on atomic theory by telling us, for example, why the elementary particle that comprise the atoms have the masses, electric charges and other physical properties that currently we can only experimentally measure.

This theory would also explain other phenomenona and physical constants that we don't currently understand, such as why the speed of light is found to be 300,000 metres per second. It would also be able to reconcile gravity with the standard model of particle physics , which we cannot yet do. A theory that explained the precise process behind the origin of the universe, and it's ultimate fate, would also be very nice to have.

As you may know, our current standard model can only deal with around four percent of the mass-energy in the universe, with the rest consisting of dark matter and dark energy that are both invisible and almost impossible to detect.

Unfortunately, this theory has not yet been developed....

But it is the role of physics to work continually towards such a theory. As part of this process, our current atomic model, which is very accurate in describing a limited set of phenomenona and physical behaviour, would be incorporated into this newer model of reality.

An example of this process, developed over the last century, is General Relativity, which explains the universe much more accurately than our previous theory, (the Newtonian Model). But we still use Newtonian physics to guide spaceprobes to land on Mars, in the same way that we can use our current atomic theory to explain Brownian motion.

We still have a lot to learn, which to me at least, is a lot better than being bored with knowing the complete theory.

Einstein's mathematical model of brownian motion furnished strong support of the atomic model but did not furnish airtight proof of its uniqueness (that is, the nonexistence of alternative models) at the time it was proposed.

It is worthwhile to note that it wasn't his objective to logically exclude the possibility of alternative models but rather to demonstrate that the atomic model furnished a quantitatively consistent explanation of the phenomenon.

His work was an important piece of a larger puzzle, contributed to by a variety of other researchers in the field, and once his piece of it was in place, the rest of the picture came into better focus- and it became much more difficult to successfully argue against the atomic hypothesis.

I think that the best answer to this question was given by Einstein in the introduction of his 1905 paper on the theory of brownian motion (the title was actually "On the movement of small particles suspended in a stationary liquid demanded by the molecular kinetic theory of heat" (which is already quite illuminating):

If the movement discussed here can actually be observed (together with
the laws relating to it that one would expect to find), then classical
thermodynamics can no longer be looked upon as applicable with
precision to bodies even of dimensions distinguishable in a microscope
: an exact determination of actual atomic dimensions is then possible.
On the other hand, had the prediction of this movement proved to be
incorrect, a weighty argument would be provided against the
molecular-kinetic conception of heat.

(from the 1956 Dover edition of the paper, translated by A.D. Cowper).

It was crystalline clear to Einstein (which did not know too much about brownian motion at the time he wrote the paper) that the validity of his theory would not had be a direct proof, but its falsification would had been a very strong case against the atomic theory. Theory that, at the beginning of twentieth century, was still considered by many mainstream physicists as an tools for computing chemical equilibria but lacking any experimental evidence.

Nothing in any science can be proved absolutely.

Atomic theory is our current best guess to explain brownian motion, among other things (such as radioactivity, all of chemistry, etc). In that sense, brownian motion supports atomic theory, but does not absolutely prove it.

One day, as StudyStudyStudy suggests, we will have a better theory, which is less wrong about the universe. However, we'll likely still use atomic theory to explain brownian motion, because it works. No need to go into quantum theory to investigate a game of pool.

That's the loop of science: theory, falsification, better theory. Because of this, we'll never reach a complete theory, and if we do, we won't know it. That's what keeps physics interesting!

EDIT: bonus philosophical argument!

How can change exist? If something changes, it's not that thing anymore, so it didn't change, so change is impossible. Early Greek philosophers saw two ways out:

a) We exist in a "block universe" where everything's one big brick and nothing happens. That's clearly not the case, so

b) The universe is made up of many tiny unchangeable bits, which make up different objects when put together in different ways — atoms!

This is just one example of falsification at work, even in the early transition from philosophy to science.