I don't believe so. If I walk into a store in New York and tell them I'm from the UK, they won't suddenly become subject to UK/EU law if they choose to serve me. I'm the one subjecting myself to US law if I shop there, not vice versa.
Further, even if they were subject to EU law on this, how would any action for a breach be enforced?
Further, even if they were subject to EU law on this, how would any action for a breach be enforced?