Well, to throw one out there, why not the Crown of Aragon? It had a large number of Mediterranean holdings, though I'm not sure how exactly to make this into a empire dominating all of the mediterranean.
The EU admits Morocco in the 80s, which causes a domino effect where North Africa, Turkey, and Israel join as well.
That is unlikely, and probably not what you're looking for (this is pre 1900 after all) but it was what came to mind first so I decided to share.
With a weaker Spain and France maybe the Ottomans could do it?
Justinian rather overstretched himself, but he came pretty damn close to reuniting the entire Mediterranean. Just add Tarragona and Marseille and you've just about got it. Could Justinian have been a bit more prudent and left his successors more money and less chaos - setting them up for more conquests?
I think this is the best scenario. Justinian's conquests were on a shoestring, but it's obvious he could have supported Belisarius better, and if he had the final cost would be far lower.
If Belisarius had a larger army, then he migth have ended the Gothic War very early, and Italy would not be devastated and be an economic benefit to the empire. Then either he or Narses could be sent to the Languedoc or Visigothic Spain.
Of course, it might still fall in a hundred years or even less, but it'd be united for at least a while.
If it's not to be the Byzantines or Muslims, it'd be a lot harder. Nothing obvious springs to mind.
Justinian probably fulfils this requirement, given that his empire controlled almost all the Mediterranean Sea.