The EU and US have an "open skies" deal, so they already "have the route".
But they'd probably face the wrath of United and the other Star Alliance TATL JV partners - without knowing what ways (if any) Aegean's hands are tied by alliance membership, they may encounter very stiff opposition to the idea of starting out on their own in competition against their alliance "partners". And if they can only do so as part of the Star TATL JV - well, then it's not quite the same thing at all.

But what they would need is the aircraft. The US is out of range of any of Aegean's fleet - even these new A321neoLRs (see Post 19 from this very thread).

Sure, they could stop off in Shannon or Dublin en-route - it would be necessary in both directions - and have their pax pre-clear USCBP on the westbound sector while refueling; or they could stop anywhere else in the western fringes of Europe for a technical stop (just to refuel) - but would that model work? I'm not so sure...much and all as I selfishly would love an Aegean flight between Ireland and the US.