In order to successfully compete in the Americas, Byzance would need to be an Atlantic power. Which means at least hanging on to the territories it had under Justinian or even to extend them. I don't think that's a realistic scenario, and in any case it would lead to a very different world from the one we know.
It also may be said that the kind of colonialisation we had IOTL was due to a competition between middle-sized nation states that led to the technological progress that made colonisation possible and made it impossible to achieve hegemony over the other nation states, so leading to these states looking for controlling trade routes and advantages in territories outside Europe. The biggest territorial empires (Austro-Hungary, Russia, China, the Ottomans) were not succesful in assembling (or didn't even try for) the kind of far-flung colonial empires that the smaller European nations created - they preferred territorially contiguous expansion. A huge Byzantine Empire (BE) as needed for an "Atlantic" scenario might very well not be interested in colonialisation.
A non-Atlantic Byzantine empire with Syria and Egyp (e.g. in a scenario without Islam, or where the BE keeps the Islamic expansion from spreading out of the Arab peninsula), would be in a similar position to the Ottoman Empire - busy with vying for influence in the Balkans, the Eastern Mediterranean and Northern Africa, the historical rivalry with Persia, profiting from its position across the pre-age-of-discovery trade routes, keeping its diverse subjects together - and therefore miss the boat on the new trade routes and decay until it's too late.
A BE consisting basically of Anatolia plus Greece (its core territories) would be a contender in the Mediterranean, but it would lack possession of the outlets out of that sea needed to take part in the colonisation game.
