A basic solution of HCl
Unfortunately, your reasoning is wrong, because you forgot to take into account the acidity of water. While in most cases we can ignore the acidity of water as the hydrogen ion concentration contribution by $\ce{HCl}$ dominates, at an extremely low concentration of $\ce{HCl},$ water becomes the main $\ce{H+}$ contributor. 

It is well known that water has a $\pu{10^{-7} M}$ concentration of hydrogen ions at $\ce{25^\circ C}.$ Therefore, water's effect on the acidity of the solution is significant when looking at solutions of $\pu{10^{-6} M}$ or lower concentrations of strong acids. For example, if there is $\pu{10^{-9} M } \ce{HCl,}$ then the total $\ce{H+}$ concentration would still nearly be $\pu{10^{-7} M}.$ Therefore, no matter how much you dilute the acid, it can never turn into a basic solution.