What's the Acid Ionization Constant of the Hydrogen Proton?
The equillibrium between a proton plus water molecule and a hydronium ion is better considered using the concept of proton affinity.

\begin{aligned}
\ce{H^+(g) &amp;-&gt; H^+(aq)} \\
\Delta E &amp;= −1530~\mathrm{kJ/mol}
\end{aligned}

You could roughly convert to an equillibrium constant using

$$\Delta G = -RT \ln(K)$$