Yes. If it is a private company they can discriminate against people with tattoos, piercings etc They can not discriminate because of skin color or religion. They do not have to hire people that they feel would be harmful to their business. Can you imagine being helped by a person with a bone in their nose while shopping for, lets say a wedding dress in an upscale bridal store? businesses have some rights left to keep their eyes on the bottom line.
"Cultural diversity" refers to race only, not bodily ornamentation. They are separate and distinct. All businesses have the right to enforce a dress code that is compatible with their business model. You won't be in corporate America with a Monroe piercing. There are, however, hair salons, boutiques and other "edgier" employers who would love someone with sophisticated tats and piercings.