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 mathoverflow][1]. 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.

  [1]: http://mathoverflow.net/questions/119495/mathematical-proof-of-the-stability-of-atoms/119528