[msk034]

I take numbers:
Proton solvation energy                             -1150kJ/m=-12.17 eV
Electron solvation energy                           -3.3 eV
Atomic hydrogen electron energy                -13.6 eV

It follows that atomic H is unstable in water and gets ionized by solvent. Is it right?

[Borek]

Numbers look interesting, but I have never seen an isolated hydrogen atom.

[Enthalpy]

Approximations:
-13.6eV is for a single atom in vacuum. The proper comparison would be with a hydrogen atom in water. Data probably impossible to find.
The entropy tells if a transformation occurs, but with a difference of 2eV, the enthalpy is good enough.

So if the dissolution of atomic hydrogen (how bizarre!) provides less than 1.9eV, yes, the atom should ionize. At least at low concentration.

This is not the last step, because water keeps the product of H₃O⁺ and OH^- concentrations constant, so some more reactions will occur.