At least for the time being, proof of a Covid-19 vaccination that has already taken place will not make it easier for you to travel within the European Union. The governments of Spain, Greece and Malta, among others, are pushing to save the 2021 summer season.
However, most of the other heads of government of the EU members cannot gain anything from this, so that the project “uniform digital vaccination pass” was put off the back burner. Commission President Ursula von der Leyen said that it was currently not at all clear whether vaccinated people could pass the virus on or not. It would also be unclear how long the protective effect lasts at all. However, the project is not entirely off the table; it is to be discussed among the EU heads of government at a later date.