I need only a valid driver’s license as medical certification to fly the more challenging of my planes
That is then the same as the UK NPPL + medical self declaration if flying below 2000kg (2000-5700kg, an AME has to check various listed conditions) and UK airspace, day VFR. Maybe the NPPL can be used in France but of course not in an N-reg due to 61.3.
I don’t think any other European country has anything similar for flying an ICAO certified aircraft.
However this huge privilege will be killed by EASA in April 2018 except for (broadly speaking) Annex 2 aircraft.
Related thread here.
f he has a bad day and decides he’s not going to pass you under Basic Med, there is never a number or FAA submission in relation to that incident and you just go to another doctor.
That is however pretty amazing!
FAA Sport Pilot is similar but not exactly the same as UK medical self declaration…. because under FAA Sport Pilot there is no declaration at all and no FAA record of that declaration. You do nothing except hold a driver’s license. However under Sport Pilot you cannot have previously failed your most recent FAA Medical, which brings the importance of the discussion above into even more focus…
It’s kind of interesting that if you’re flying under Sport Pilot the FAA has no idea if you’re still out there flying or not, the periodic medical filing being a private pilots only interaction with FAA. Maybe after you reach 200 years old in the FAA pilot record they’ll remove you.
Re Basic Med you can imagine all the recent hangar talk among pilots, dissecting it and figuring out the various angles. Worse than an homeowner’s association meeting in an affluent neighborhood 
