The size of the Roman Empire at its height?! You're going to need an early POD. Firstly, merchant republics aren't built for maintaining huge empires like that. See Genoa, Venice, the Dutch Empre, etc. They mostly just dabble wherever an economic opportunity is to be made.
Anyway, here's how I would go about doing it; some time in the seventh century the Lombards in Northern Italy leave Milan for some reason and focus on Venice. They mainy focus on naval matters, and over the next few decades they have conquered much of the Adriatic coasts. The Venetian Kingdom allies with Byzantium and take the rest of Italy and Tunisia from the Italian kingdoms and the Arabs. After another couple of wars with the Caliphate, the Venitians have taken Iberia and the rest of the Barbary Coast. They absorb Gaul and establish trading intrests in Englalund, which gradually turn into full-fledged colonies. By the eleventh and twelvth centuries, Venice has taken the rest of the Balkans and Egypt, and some time around 1200 Venice and Byzantium form a union. Over the next century this union solidifies and by the fourteenth century the Venetian Empire rules the Mediterranean. Not very plausible, but it's the best I got.