Yes, and it's doing a mediocre-at-best job of it. You can remove tariffs and harmonise regulations, but the EU is trying to unite countries with cultural borders older than Christianity.
Obviously there is a lot to criticize about the EU and I can offer you a gigantic list there too. However, I do not see any clear failure of the EU’s approach as a single market so far. Additionally part of the philosophy was establishing peace in a region that was torn up by wars for a lot longer than Christianity exists. I would argue the EU was quite successful there too.