Which of these laws is more fundamental or forms the basis of electrostatics? I started off with Coulomb's law and then I studied Gauss' law. I was wondering which one is more universal?
My professor derived Gauss' law using Coulomb's law but didn't do it the other way, so is Coulomb's law more fundamental? And can Gauss' law be used to prove the other?
Because Gauss's law applies for both moving and stationary charges, while Coulomb's law applies only for stationary charges, Gauss's law can be considered more fundamental. This is why Gauss's law is one of the four Maxwell equations. The derivation of Gauss's law from Coulomb's law only works for stationary charges; for moving charges the derivation is invalid yet Gauss's law still holds. However, Gauss's law along with the information from Maxwell's third equation that the $curl E = 0$ for stationary charges (since then $B$ will be constant), can be used to derive Coulomb's equation. In short, Gauss's law can be considered more fundamental because it applies to both stationary and moving charges, while Coulomb's law applies only to stationary charges.
If we are only considering three spatial dimensions, then Coulomb's and Gauss's laws are completely mathematically equivalent and there is no basis to consider either to be more fundamental than the other. But in several dimensions other than three, they are no longer equivalent, and when theorists consider generalizing electromagnetism to other numbers of dimensions, they almost always keep Gauss's law the same and modify Coulomb's law. So in that very weak sense, one could consider Gauss's law to be more fundamental. I discuss here why they do so. At the end of the day it boils down to a philosophical preference for mathematical elegance; until we find a universe with a different number of dimensions, there is no "right" answer.
From the Feynman Lectures on Physics (I would have made this a comment but I don't have enough points)
From our derivation you see that Gauss' law follows from the fact that the exponent in Coulomb's law is exactly two. A $1/r^3$ field, or any $1/r^n$ field with $n≠2$, would not give Gauss' law. So Gauss' law is just an expression, in a different form, of the Coulomb law of forces between two charges. In fact, working back from Gauss' law, you can derive Coulomb's law. The two are quite equivalent so long as we keep in mind the rule that the forces between charges are radial.
