Crop rotation, green manure and compost are quite irrelevant in hydroponics and aeroponics, and the "fertilizer" is synthesized too, no?
This is debatable. So there is no clear answer and lots of opinions. You can theoretically derive hydroponic fertilizer from 'organic' sources and not synthetically produce them (from chemical rather than biological processes). But whether this is an organic method is debatable. That is: Is growing plants without a biological soil, an 'organic' method? There are lots of commercial 'organic' hydroponic nutrient solutions out there. But these types of products are part of an industry rife with hyperbole, misinformation, and outright fraud. Making your own from locally derived sources is possible, but you really need to know what you are doing.
Never tried any ponics but it seems to me like you could make a nutrient solution fairly easily by washing compost, filtering the solids using an appropriate filter and sterilizing the liquid for any pathogen that got in.
It should work given that's what plants consume in normal conditions.
It might be hard to get consistent results of nutrient balance and PH without advanced equipment though.
Probably none of the agencies give your products organic documents, but yes you can do it. If you prepare fusion with organic stinging nettle and a few other kinds of green plants you can grow organic in aeroponics. The point is you should provide more than enough oxygen in fusion water continuously otherwise your nutritious water solution will be toxic and provide a rich environment for unwanted bacteria and yeast.
I think it depends on your definition of organic - different agencies have different definitions. Some even allow the use of some inorganic (as in a chemist's definition of inorganic) salts. Is it natural no, that's not how the plants grow naturally, but then vegetable plots are not natural habitats and most vegetables are highly cultivated and cannot be considered natural. Can you use plant or animal derived nutrients like you can in the garden, yes. Can it be considered ecologically sound, depending on set up, yes, aquaponic systems could be considered particularly low-impact. My opinion, as no definitive answer is possible, is that they can be "organic" whatever that means. As an occasional chemist, the word irritates me - to a chemist all plants are organic.