It isn't clear yet as it is only a proposal at the moment but I'm guessing it would require full vaccination as you aren't properly vaccinated until you've had both doses (or one in the case of Johnson & Johnson). You may also be able to travel if you can prove you have had covid. This is how it works with Israel's domestic vaccine passport (green pass).
It's also not clear whether all EU countries would be required to use a vaccine passport and there are numerous other issues to be worked out. For example Hungary has been using Sputnik and Sinopharm vaccines which aren't authorised by the EMA yet so those people wouldn't be covered by the passport.
Then there is the question of whether we can come to an agreement with the EU to be covered by the passport. In theory it shouldn't be a problem as we are using the same vaccines they are but again who knows.
It will all become clearer soon enough. Just don't book a holiday unless you know you can get your money back or are willing to lose it as we don't even know if we will be able to travel at all, let alone make use of vaccine passports.
From the bits I've seen it sounds like the rules regarding International Travel will become clearer next month. In any case I imagine that people will definitely be flying to places by June/July in any case, but imagine there will just be a list of blacklist countries/locations that you have to quarantine on return from.
I think there will also likely be some form of requirement for Proof of Vaccine/Negative Test Result.