F9 was designed with one eye on recovery, Ariane 5 wasn't. It would be a more or less new design if ArianeSpace went down that road. Can you even restart the Vulcain 2?
Not as it is. It needs a lot of conditioning for ignition, after all, it is a large cryogenic engine.
A Vulcain III with the necessary modifications would be possible, but not instantly done. For example, you need to keep many components in the right temperature box for not damaging them when the engine starts or stops.
The better question though is: Is it neccessary? I just remind on the Phoenix Hopper plans some years ago. Contrary to SpaceX and the USA in general, Europe has the means to let a stage land after an aerodynamic glide on a runway in the east Atlantic. Instead of using rocket engines and vertical landings, there would be a horizontal landing.
It would need a more complex landing gear, but at the same time, this landing gear could be easily acquired commercially off the shelf - cost reduction. Since a stage is very large and very light, it would not require an overly sophisticated landing gear like the Space Shuttle, anything for a larger Dassault business jet would be good enough and the same ground infrastructure would be needed. The structural mass for wings and the fact that the static loads vary largely during the mission profile is a bigger problem.
The know-how and the software for such landings exists already in Europe and within Airbus Space, we don't need to start at zero.
A Merlin "IIb" with only a short phase between burns would be no problem to create, and you could also do such a mission profile as single burn with an existing Merlin II.
On the other hand, when discussing such plans - we could also put our eggs into the Skylon basket. Which likely has the chances to let SpaceX look old. It could be far more economic, if the process planning holds what it promises.