What is the best way to prove Aristotle's theory of when a plant grows all material comes from soil to be incorrect or correct?

What you are calling a theory is based on a single testable observation. That makes it an hypothesis. A theory wants to collect up a variety of related hypotheses and derive a grand answer to why a system works the way it does.

An hypothesis must be testable by empirical means and it must be provable false. If you fail to prove it false, you can accept it. That doesn't make it true, though.

Now that we've reduced Aristotle's contention to an hypothesis, you must recognize that an hypothesis can never proven true. You can only support it. You can prove the hypothesis "A plant gains weight by obtaining material from the soil." false. if you weigh the dry soil before and after and find that the weight didn't change, the hypothesis is false.

If the weight after was the same, you might postulate that the plant is getting its mass from the air and the water (new hypothesis). If you repeat the experiment a number of times and it always results in the same result, that gives greater support but it never makes it true.