The electrolyte (sulphuric acid diluted with distilled water) in a lead/acid battery is not there to conduct electricity. It's function is to produce a chemical reaction with the lead plates in the battery to produce electricity.
Distilled water is used because the minerals in tap water would increase the speed at which the lead plates become covered with lead sulphate which would prevent the chemical reaction taking place. (the acid has to be in contact with lead not lead sulphate)
The dangerous gas seaangler refers to is hydrogen and hydrogen sulphide which is a natural product of the charging process of a lead/acid battery.
Here endeth the first chemistry lesson.