You need H+ to raise the pH, like when you split water in HO- and H+. [1]

In this case they get H2O2 (neutral) so the H must be neutral, so they must combine to form H2 (neutral). (Perhaps the H radical can survive some time in water until it combine with other H and form the H2.)

[1] A small technical detail is that the reaction is 2 * H2O --> H3O+ + HO-, not H2O --> H+ + HO- , but H+ is the usual shorthand.