Old English bæð "an immersing of the body in water, mud, etc.," also "a quantity of water, etc., for bathing," from Proto-Germanic *badan (source also of Old Frisian beth, Old Saxon bath, Old Norse bað, Middle Dutch bat, German Bad), from PIE root *bhē- "to warm" + *-thuz, Germanic suffix indicating "act, process, condition" (as in birth, death). The etymological sense is of heating, not immersing.
The city in Somerset, England (Old English Baðun) was so called from its hot springs. Bath salts attested from 1875 (Dr. Julius Braun, "On the Curative Effects of Baths and Waters"). Bath-house is from 1705; bath-towel is from 1958.