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 is attested from 1875 (Dr. Julius Braun, "On the Curative Effects of Baths and Waters"). Bath-house is from 1705; bath-towel is from 1958.
c. 1300, "vessel for cooking," from stew (v.). Later "heated room," especially for bathing (late 14c.). The meaning "stewed meat with vegetables" is first recorded 1756. The obsolete slang meaning "brothel" (mid-14c., usually plural, stews) is from a parallel sense of "public bath house" (mid-14c.), carried over from Old French estuve "bath, bath house; bawdy house," reflecting the reputation of medieval bath houses.