If it were possible for one object to pass through another object, then it would be possible for one part of an object to pass through a different part of the same object. Therefore the question asked here is equivalent to the question of why matter is stable. See this question on mathoverflowthis question on mathoverflow. That question was more about the stability of individual atoms, but in my answer there, I gave a reference to a paper by Lieb. Section II discusses the stability of bulk matter. The argument depends on both the properties of the electromagnetic interaction and on the Pauli exclusion principle. Therefore anyone who tells you that the stability of bulk matter is purely due to one or the other of these factors is wrong.
The same holds for the normal force. People will try to argue that it's only due to electromagnetic interactions or only due to the exclusion principle. That's wrong for the same reasons.
Although Lieb's treatment is complicated, there is a pretty straightforward argument that quantum mechanics is necessary for the stability of matter. There is a theorem called Earnshaw's theorem that says that a classical system of interacting charged particles can't have a stable, static equilibrium. It's not a difficult or deep result; it's just an application of Gauss's law. Letting the equilibrium be dynamical rather than static doesn't help, since then the charges would radiate.
