I am sorry, as i do have a small Motorcycle (Hobby) Shop, Metal polish is something i think of as a "home remedy"As i think the stuff is only on the backside, a few hair scratches won´t do no harm.You can try Aluminium foil , crushed to a ball, as a scrapüer, combined with some water and dishwashing liquid.I would advise against salt and vinegar as long as you do not know what metals are used exactly, you could accelerate corrosion.
I guess this watch has a brass alloy casing which is chrome or nickel plated. Sweat will cause small pores in the plating and the base metals oxidation/corrosion products will form teh builup.