Is it legal to turn high beams on when driving on a European highway with fully separated lanes?

Because you specifically mentioned the Autobahn, I explain the situation in Germany.

The relevant passage is § 17 der Straßenverkehrsordnung über Beleuchtung, Absatz (2):

Auf Straßen mit durchgehender, ausreichender Beleuchtung darf auch nicht mit Fernlicht gefahren werden. (On roads with continous and sufficient illumation high beams must not be used).

The Bußgeldkatalog mentions the fines for this offense: From 10€ to 35€ (117130 to 117132) depending if nothing happened, you endanger other people or caused an accident.

The current ruling is that in fact fully separated autobahns permit the use of high beams on unilluminated parts if the conditions suggest it: Ist der Mittelstreifen auf der Autobahn ausreichend lichtdicht und werden andere Verkehrsteilnehmer nicht geblendet, darf mit Fernlicht auf Autobahnen gefahren werden.(In case that the road center is sufficient opaque and other persons are not blinded, high beams may be used on autobahns).. This means night or other low-light conditions (thunderstorm). In all cases you still should immediately turn high beam off if someone is ahead of you or oncoming traffic is able to see your lights.

If you use high beams if you are alone on an unilluminated autobahn under low-light conditions you should be fine. For fog and strong rain (sight under 50m) always use fog-light and don't forget to switch it off after leaving the fog bank.

A sincere warning: Do not use high beams near deer crossings.

Public domain: Bundesanstalt für Straßenwesen, 2016-12-15

If you are using high beams, wild animals will stop and stare hypnotically into your light and given the high speeds cause severe accidents. There are always some warnings in German travel radio if deer has been sighted on the autobahn.

ADDITION: Makyen commented correctly that game could be caught in low beams. But this happens more on country roads (Landstraße) because they are difficult to observe: curves, the wood is very near the road and you are slower. On autobahns however, the situation is different and high beams are more dangerous: Autobahns are straight or only slightly curved, they are broad (at least 2 lanes), there is always a big margin between wood and road and you are moving with speeds of 130 kph (80 mph, 36 m/s). Game could not effectively "hypnotized" in low beams because those have a range of approx. 50 m and this means collision will be almost inevitable. It is also rare because game can hear cars at such speeds and they will either retreat hastily or even if they run directly into your vehicle, chances are good that they do not match exactly the speed to get into the front, so a glancing blow is much more likely. High beams on the other hand will catch game outside hearing range and "freeze" them directly ahead of you. In this case lowering the beams besides braking and honking is the correct reaction, so lowering the beams beforehand is a good strategy.

I live in Denmark, and yes, it's legal to use high beams on the freeway

There was recently a radio show that discussed the use of high beams on Danish freeways, here's a (translated) quote from the interviewed police officer:

Yes, it's legal to use high beams on the freeway, and it's encouraged to do so. As long as your own side of the road is clear (ie. nobody in front of you), you are perfectly allowed to use high beams.

If there's oncoming traffic, you don't strictly have to turn down your lights. The risk of blinding others is outweighed by the need of proper vision at freeway speeds (110-130 km/h in Denmark), and the light is usually blocked by bushes or road guards

Of course it does come with usage of common sense, and it is the judgement of the driver himself that determines whether or not high beams are required.

The legal stance of this varies from country to country... in some places it's illegal (mainly due to deer, elk orother wildlife that get"tranced" by the lights)

I (and many other Europeans) never drive with high beams on the freeway. Often the nights are not that dark (especially in summer or moonlit nights), so you can still see quite a distance ahead, provided your night vision is ok.