Why would the loss of H₂ to space mean that the Earth's geochemistry would be based on acid/base reactions?
The statement you cite contains a reference to “How Earth’s atmosphere evolved to an oxic state: A status report”, D. C. Catling and M. W. Claire, Earth and Planetary Science Letters 237 (2005) 1–20. This paper in turns give some more details on this assertion, which I'll summarize below.

It all turns around the question of how reducing the prebiotic atmosphere was. The paper I cited above contains this quote:


  before life existed the atmosphere contained mainly $\ce{N2}$, with negligible $\ce{O2}$, probably more $\ce{CO2}$ than today, and only minor levels of reducing gases such as $\ce{H2}$.


So, in the absence of widely available gases such as oxidative $\ce{O2}$ or reducing $\ce{H2}$, oxygen was mainly present as $\ce{H2O}$ and $\ce{CO2}$. This means redox reactions are not big players in the overall chemical scheme of the periodic atmosphere, and acide-base reactions dominate (by default).