Or maybe that is unrealistic.
Even with a maintained Roman imperium, it would be hard to unify Europe as a continent (or even to consider Romania as one homogenic block, culturally or politically).
Of course, the big disinterest for anything happening deep on the Barbaricum would lead to ignore Viking-like discoveries in New World, so it may be irrelevant.
As for along the Chinese lines : before the Mongol takover, I'm under the impression that China was quite active in maritime matters past the Sea of China (hence T'ang traders in India) beneficing as well from cultural tropes about being the center of the world and expecting other nations to go to China rather than the reverse.
Roman mentality, especially in the latter time, was roughly similar (it was admittedly built on much rationalisation) but had a definitely more important interventionist trend. Expect a cultural shift, I don't see why Romans (in the large sense, including creolized populations) would ignore the growing African market and wouldn't open up to Atlantic with having the ressources of a whole Meditteranean basin and most of Europe behind them).
The main difference would probably be not if they would discover Americas if they survived, but how they'd have done afterwards.
Giving the likely maintain of slavery (whatever in latifundian or villae production system) and the need to rentabilize discoveries, it wouldn't be pretty but you could avoid continental takeover more easily due to other fronts (Barbaricum, Persia, Africa), while still considering the really important ressources at hand.