A question about acidity
Not really. Like lots of things with polar bonds, sulfuric acid can undergo ion transfer reactions in the liquid and make ionic species in solution.  And it is indeed protons that get transferred just like water.  But as with water, at least under ordinary conditions, these ionic species are only a minor portion of the liquid which is mostly $\ce{H2SO4}$ molecules.

Strong acids such as sulfuric acid are indeed acidic, capable of acting as acids.  But they still need a base with which they can react, to actually act as acids.  In the case of diluting sulfuric acid, water serves as such a base.