Chemistry teacher here! Hydrogen would only acidify it if it dissociated. Much like how you can dissolve oxygen or nitrogen gas into water, any gas can be dissolved into water. They don't break apart, they just float as molecules inside the water. It's just like when sugar dissolves. Salt breaks apart, because it's ionic. Sugar, most organics, and diatomic gases like H~2~, N~2~, and O~2~ don't have enough affinity with the water molecules to dissociate (or, at least, not sufficient to dissociate appreciably)

When you get something gnarly is if you have a molecule containing something that does have stronger affinity with the water. Carbon Dioxide, Sulfur Di- and Trioxide, Nitrogen Oxides, and other oxygen-bearing covalent gases react with water because the central atoms attract the oxygen in the H~2~O, while the oxygens surrounding them have partial negative charges from the unpaired electrons, attracting the hydrogens in the water. This causes the water to be ripped apart, creates oxyanions such as CO~3~^2-^, SO~3~^2-^, SO~4~^2-^, NO~2~^-^, or NO~3~^-^, and leaves protons in the water from the now-dissociated Hydrogens (except for weak acids such as Carbonic Acid, which only partly dissociate from the hydrogen, such that all intermediate species are actually in equilibrium: H~2~CO~3~, HCO~3~^-^, CO~3~^2-^, and CO~2~). Same goes for elemental Chlorine, Fluorine and Bromine. All of these rip the water apart and create the hypo- oxyacid and the hydroacid of the specie (e.g. Cl~2~ + H~2~O --> HClO + HCl)