Hmm, I would like to argue that H2SO4 is a strong acid, stronger than HCl (Not that comparative strength of strong acids are needed for the HSC, but regardless)
To quote both steps of ionisation of H2SO4 as the strength of the acidity of H2SO4 is not completely right IMO, since, by LB theory of acid and bases, acid merely donates one proton. This implies that the strength of H2SO4 should be based on the below ionisation:
H2SO4 + H2O ---> H3O + HSO4 -
This ionisation step, IIRC, is very strong (stronger than the ionisation of HCl).
Now that been said, the strength of HSO4- is alot weaker than HCl (as a lot of you have mentioned already).
edit: #minijunk: on the topic of complete ionisation, it's a over simplification. No reaction in the universe is complete. Just at times, the equilibrium constant is so huge (such as for first step of sulfuric acid ionisation), that for all practical purposes, it can be treated as infinite (i.e. complete reaction)