NO... and YES.
Technically (and legally) speaking: NO... and if you happen to get entangled in a serious "incident" (a brawl, an accident, a terrorist alert...) and you're asked by the Police to produce your ID and have no valid ID (a photocopy is not legally a valid ID!), you are very likely to be taken to the Police Station until someone can bring your passport for you. Not necessarily a frightening experience (don't worry, Catalan Police is far more civilised than the average US cop, in principle, they never cuff you unless they think you're a menace... and they normally "ask" you to accompany them to the station), but a waste of your limited vacation time for certain. The Law states EVERYBODY must carry a legal form of ID at ALL TIMES. And for you, that's the passport, as mentioned earlier.
In practice: sort of YES... for most "normal" situations (excluding the above) that is, say, for example, to prove your age if you bought reduced tickets in a museum. But again, it wouldn't be accepted if paying by credit card in certain stores (or in supermarkets). Also, the vast majority of restaurants don't ask for ID for a "normal" check, yet if you are dining in a fancy restaurant and run a bill of 500€ then they might want to ensure it's your credit card that you're using, LOL!... and it's their legal right to ask for ID when paying with a credit card so they are entitled not to accept a photocopy.
Unfortunately, there are some people and some websites out there propagating this myth about "a photocopy is enough"... but I am afraid that for ID purposes, the Law says otherwise, and not only in Spain, also in some other European countries. This is not to say you might "get by" in certain circumstances with a photocopy, but again... you only need a driver's license if caught by the Police, otherwise nothing impedes you to drive a car, right? :))