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.
early 12c., Nativite, "feast-day celebrating the birth of Christ, Christmas," from Old French nativité "birth, origin, descent; birthday; Christmas" (12c.), from Late Latin nativitatem (nominative nativitas) "birth," from Latin nativus "born, native" (see native (adj.)). Late Old English had nativiteð, from earlier Old French nativited. From late 14c. as "fact of being born; circumstances attending one's birth."