There isn’t any “official” Church document with regards to the EU in the sense of some infallible doctrinal pronouncement.
If you want to see some of what has been said, you might just do a Google site search of the Vatican website for “European Union.” Here are some of the top hits:
The Church is generally in favor of bringing people together for dialogue, particularly considering some of the devastating effects of nationalism on the continent during the 20th century. So certainly they are supportive in that sense. That is a far different thing from the Vatican endorsing and approving everything the EU says and does. The Vatican has never done that.
I think you’ll find that COMECE, the Commission of the Bishops’ Conferences of the European Union, has the answers to all your questions on this subject. http://www.comece.eu/site/en/whoweare
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