No, I was answering your question specifically:
My question is, can we consider that this species has become naturalized in France?
The CABI database is usually a good place to check the status of non-native species, here is the entry for Psittacula krameri specifically, with the portion concerning France quoted below:
In France, P. krameri has become established in Aix-en-Provence (introduced 1999), Marseille (introduced 1996), Ares, Nancy (introduced 2005), Villeneuve-d’Asque (introduced 2002), Frejus (introduced 1999), La Garde (introduced 2001), L’Isle-sur-Sorgues (introduced 2002), Wissous (introduced 1984), Drancy (introduced 1990), Nogent-sur-Marne (introduced 1993), Jagny-sous-Bois (introduced 1970), Lille (introduced 1990) and Strasbourg (introduced 1999). There were populations, now extinct, in La Roche de Glun (introduced 1983) and Saint-Gilles (introduced 1990). The species has been introduced but the fate of the population is unclear at the moment in Cannes (introduced 2003), Sigean (introduced 2003), Montpellier (introduced 2007) and Mercues (introduced 2007) (Strubbe and Matthysen, 2009a).
Given that it’s listed as reproducing on its own in Belgium and Germany, it’s safe to conclude that it is doing the same in France.
The IUCN GISD database is another good reference source, here’s the specific species entry:
There is is listed as not just naturalized in France, but as actively invasive.
So, yes, it should be considered naturalized, furthermore it should be listed as invasive, at least according to the agencies that monitor this sort of thing.
I’m sure there is a French government site that keeps track of this sort of thing as there is in the US, Japan, Germany, Finland, and many other nations. Likely ones maintained by each province as well, as is also common. See if you can track those down and see what French government sources say on the subject.
The EU also tracks this sort of thing more broadly, here’s the result of a quick search for the species on the European Commission site, take a look through the 10 papers there and see what they indicate: