There are a few points relevant to this "European Aircraft Carrier":
- In February 2003 at the Le Touquet Anglo-French summit Jaques Chirac and Tony Blair agreed to work towards a pooled carrier capability for supporting the then-planned European RRF. It was planned to involve other European nations in this, in particular Italy and Spain with existing capabilities. That's the basic idea behind the whole thing.
- At the time the defense spokesman of the CDU publicly endorsed the idea of a pooled/shared European carrier capability and endorsed that Germany "should join such an initiative". The same guy went on to stay as a parliamentary secretary (lower-rank minister) at the German MoD until 2013 and then was Federal Minister of Agriculture and later Federal Minister of Transport in Merkel's cabinets - in other words we're talking about a defense spokesman deep in Merkel's circles. He was kicked off the government bench with the new coalition a few months ago.
- The primary opponent of the above plan decided to quit Europe in June 2016.
- France is - since October 2018 - running a study to evaluate future requirements for a successor to Charles de Gaulle, to commission in the timeframe of 2030-2035. Set requirements include compatibility with the Franco-German Rafale/Eurofighter successor. According to a statement of the French Minister of the Armed Forces, part of the studies is evaluating "how many carriers will be required for French and European interests".
And then after that comes the Treaty of Aachen.
It will be interesting to see what eventuates with this. If it does go ahead, will it include Italy and Spain, both of whom have carrier experience especially with VTOL aircraft, to whit the Harrier, and Italy is migrating to the F-35B subject to political shenanigans and convulsions etc., in Rome. Spain may or may not migrate to the F-35B because of funding. If both of them do, then that is a powerful force multiplier for a Euro CV force.
That's the next question, do you build one or two CATOBAR CVs? Or do you build one CATOBAR CV and one STOVL CV? It does seem pointless Italy and Spain building a STOVL CV each when they and EU Defence could pool resources and build one, possibly even two. Two STOLV CV and one large CATOBAR CV or CVN would give EuroDEF quite a potent naval force. If perfidious Albion, when it isn't undergoing self inflicted political shenanigans, ructions, and self immolation e.g., were to join said force on occasion that makes for a very strong force that would make even those across the pond sit up and take notice. Would there be political will within Europe to build a CV force consisting of one CATOBAR CVN and two STOVL CVs? There are enough capable ships in all the Euro navies to provide escorts and a logistics train for a Euro CBG and such a CBG would be quite effective.