Damaged (scraped) tire - what to do?
It does not look to me like the cords are damaged, which means the tire will still have its strength to hold the air pressure of the tube. 

You have three easy options. Personally I would take the last:


Ignore it. It will probably last until the tire has no tread left. The cut itself will bulge a little, so this will be the point of failure of the tire. The bump may cause the tire to make noise of vibration as you ride- If you notice this the tire has little life left, or may quietly wear down till with the rest of the tire. 
Put a sleeve in. Get a piece of tough material like canvas (the side wall of an old MTB tire is great), insert it under the cut. This will provide a stronger area around the cut and prevent the weakness causing further problems and extend the life a little more as it will reduce the size of the bulge caused by the cut. Its a technique used as a temporary repair for tires to get you home, and works for tires with significant cuts (I have seen it used successfully on a MTB ride for a tire with a 2-3cm full thickness cut.)  As we all know, there is nothing more permanent than a temporary fix that works. 
Replace the tire. A shredded tire on the road is really a pain. Unlike a puncture, which is generally repairable, the only fix is a new tire - do you carry a spare tire? Although I always carry a tube and repair kit, I only carry a spare tire for very long rides (and usually just one in our group will carry a spare). If I have a spare tire in good nick, why would I leave a dodgy tire on the bike?