Unlike, say, sending a rocket to orbit, there's not a particular physical threshold to crossing the Atlantic. You mostly just have to float. Pretty much any ship can complete a transatlantic voyage if you know what you're doing and/or you get lucky. Thor Heyerdahl in 1970 crossed the Atlantic in a papyrus boat named Ra II, meant to simulate ancient Egyptian craft. Other modern people have crossed the Atlantic in small sail boats, solo rowboats, log rafts, barrels, and even with a kickboard. So it's plausible that a Roman ship could make it... or for that matter an Irish saint in a currach.

But that's a really big can in my statement above. You'll note the numeral in the name of the Ra II; Heyerdahl didn't make it in the original Ra (though he did get surprisingly close.) Storms and leaks sunk his first try a hundred miles short of the Carribean. He was rescued, but of course real Egyptian sailors would not have had that luxury and would simply never have been heard from again.

The Atlantic commonly develops forces that would make all these modern crossing stunts impossible if they encountered them, so they required both good planning and good luck. An ancient Mediterranean ship could handle some amount of wind and waves; they were part of a very long sea-faring tradition and were hardly a bundle of matchsticks. They did indeed navigate the coasts of the Atlantic, and even out to the Canaries. But they could easily get overwhelmed in the Atlantic. They were fairly lightly built with low freeboard. The sail arrangement didn't permit much in the way of sailing into the wind, and with only one or two main sails it could be difficult to adjust to the high variations in winds. The control authority of the steering oar gets quickly overwhelmed in high seas.

It also takes quite a while to cross the Atlantic under sail, and you need to have enough provisions, especially water, so that you don't die before you get there. And Roman ships run into some trouble here; they're slow. If a Roman sailing ship had a top speed of 6 knots, as their very fastest may have, that's something like a 30 day crossing in good conditions. With notable poor or adverse winds, that gets worse. A rowed galley may have a faster top speed, but can't keep it up all day and burns a lot of calories and water in doing so. A recent two-person rowed boat made it in 45 days.

A Roman cargo sailing vessel could certainly have laid in a couple months supplies, had they known they needed to. A bireme would have to be absolutely stacked to the non-existent rafters with food and water.

So if the question is instead "did the Romans or Greeks have ships that could confidently and reliably survive an Atlantic crossing", the answer is "not really, but you can try if you have a high risk tolerance."

All this, of course, focuses on vessel capability, and there's a lot more to actually doing it than the ship. By far the most important tool in crossing the Atlantic is knowledge. Global wind and weather patterns, ocean currents, cartography, navigational positioning, and, frankly, the concept that there's something there on the other side are all rather important to successfully crossing the Atlantic, especially in marginal conditions, and were all rudimentary or missing in classical antiquity. And those concepts were all much better known by 1492, when the Spanish and Portugese had been doing the "volta do mar" using the trade winds to go well out to sea to transit to (and discover) Madeira and the Azores via the Canaries. (Of course Columbus was wrong about what was on the other side, but he needed that concept to make the attempt.)

As an aside: Could the Romans or Phoenicians have known how to do the volta do mar? They very definitely visited the Canaries, but those are visible from Africa and you can row there. They may have heard about Madeira, but did not know of the Azores. So if they were taking advantage of the trade winds, they weren't doing it much.

So I'd say in summary, they could have made it, but it would have required good luck to make it there... and back. You'd have to catch decent winds both ways, know you needed a lot of supplies, missed major storms and hurricanes, not developed any serious leaks for several months straight, and a lot more. Columbus as an example had pretty good weather luck, but only his third expedition of five didn't encounter any severe weather. Possibly a Roman expedition in the same circumstances could have survived the storms on the return of the first voyage, or the odd storm (possibly a tornado) while in harbor on Hispaniola during the second, but it seems unlikely they'd survive Columbus' fourth or fifth voyages when he barely did.