The big question about capability here isn't actually the seaworthiness of the vessels per se - Roman and Greek vessels in classical antiquity could be quite well engineered for specific purposes. We know from the shipwrecks of larger vessels, with dead weight tonnages between 300 and 600 tonnes, as well as surviving primary sources, that ancient shipwrights knew a fairly decent amount about hydrostatic principles. Were they incentivised, at any cost, to specifically design a ship for open ocean, they almost certainly would have been able to construct a vessel that, physically, could have survived the voyage across the Atlantic.

The key issue of debate here is navigation and knowledge of the prevailing wind patterns of the Atlantic. The first Europeans to cross the Atlantic, Vikings, did so without compasses, but they possesed significant knowledge of the prevailing winds of the North Atlantic and were able to somewhat reliably determine latitude, particularly by using handheld 'sun compasses' to measure the shadow curve of the sun.

With the Roman and Ancient Greeks the big issue is from all we know - they didn't possess the navigational capabilities for an Atlantic voyage. By and large their methods of maritime navigation were based around Mediterranean voyages.

In my, personal opinion, the earliest European societies which could have built a seaworthy enough vessel to cross the Atlantic are several of the Hellenistic kingdoms, and the Roman Republic. The earliest actually capable of navigating the voyage are the Viking seafarers.

Sources-

Alain Bresson, The Making of the Ancient Greek Economy. Princeton University Press, 2015. Of particular interest are his chapters on maritime commerce and shipping in both the Hellenistic and Roman periods.

Paddy Griffith, The Viking Art of War. Greenhill Publishing, 1998.

Stefan Brink and Neil Price, The Viking World. Routledge, 2015.