The Poll is with a POD of 1400 (we can push back 50 years or push up 50 years depending upon circumstance) and with an ending date of 1700 (the hegemony over Europe or the Mediterranean must be completed by 1700, so no Napoleonic era styled situations), to choose among those in the poll, which country was most likely to either:
1. Form an hegemony across Europe. That is, either they rule indirectly through a series of vassals or tributaries all of Europe or they rule these lands directly. Whichever is most likely. Defining Europe: we refer to the lands beginning in Portugal and ending at the Kingdom of Poland, thus we refer to a limited Europe; Europe as in the Latin world if you will.
2. Form an hegemony across the Mediterranean. Namely, that said power is able to make clients, vassals or directly rule the entirety of the sea, in a manner resembling the Roman Empire of 120 CE.
Note, exceptions can be made by powers. For instance, the Papacy can achieve this goal if every monarch submits to Papal decrees regarding acceptance of Papal vassalage.
Then, let’s start with the list of the exclusions from your poll: Poland, Lithuania, Novgorod, Moscow and the GH are out because none of them was anywhere close to the Mediterranean and, with the exception of the GH and, for a short while Lithuania, around 1400 none of them had access even to the Black Sea. Well, Kalmar Union also was not anywhere close geographically and the same goes for England which was at that time far away from “ruling over the waves”.
Out of the rest,
in the short term probably the Ottomans. Even with the offset caused by Timur, they still have the best military system in the Med region (of course, it took time to come back).
In 1400 there was no Charles V/Carlos I or even united Spain.
Aragon had a fleet but did it have a strong army circa 1400?
France was between Caroline (ended in 1389) and Lancastrian (started in 1415) phases of the 100YW but hardly in a good position for the expansion across the Med (among other problems, its king started showing signs of madness in 1392 and there was an ongoing quarrel between the Orlean and Burgundian parties).
The Papacy was in the midst of the Great Schism (1378 - 1417).
The HRE between 1346 and 1437 was under the House of Luxembourg and Sigismund was just beaten by the Ottomans at Nicopolis in 1396 and had to deal with other numerous problems (among them the Hussite Wars which started in 1419). In Hungary he was imprisoned once and deposed twice and then had problems in Italy with Laduslaus of Naples. As far as the Med (or rather Adriatic’s) goes, he got possession of Croatia and sovereignty over Serbia but he was also busy with the Polish-Teutonic quarrel, Council of Constance and the Husittes. Hardly a good situation for the serious expansion on the Med.
Byzantine Empire by 1400 was almost dead.
Venice had limited military resources which would not allow to go beyond possession of the islands on the Med.
Did Mamelukes have a powerful navy
and stable dynasty?
But
in a long term for the Med we can probably have competition between Spain (in the broad terms), France and the Ottomans. If you do not insist upon the complete territorial possession, probably by 1700 France may become the #1 candidate if it concentrated on this specific area instead of trying to run in all directions simultaneously.
Starting from the reign of Louis XI it should concentrate on building the “modern” army based upon the Swiss model augmented by powerful artillery (which it had by the time of the Italian Wars) and backed up by the traditional armored cavalry. Infantry has to be initially Swiss but creation of the national units had to be started immediately. Introduction of infantry firearms should happen as soon as they became practical. This could make all the difference in the Italian Wars and, being better economically developed than Spain and having a greater population, France would be able to modernize its army with a greater ease (which did happen). Preferably, the Wars of Religion are avoided.
The same goes for the earlier development of a navy with the initial heavy reliance upon the Genovese but steadily growth of the French own navy and early moving from the galleys to the sail and artillery ships.
France should be able to maintain presence in part of Italy (Milan, Genoa, Naples, perhaps Piedmont) and dominate the rest. With a fleet strong enough (and alliance with the Ottomans) it could end up with a possession of some important islands. Avoidance of the military conflicts in Germany would allow to concentrate resources on the Med containing Spain and eventually making the Ottomans a junior partner in anti-Hapsburg scenario: in an absence of the French aggression in Germany, the Hapsburgs are on their own and if France is actively supporting the Dutch in their wars with England, then the British penetration into the Med is also limited.
I’m not sure that European dominance by a single power by 1700 was possible. Of course, it can be argued that LXIV came close because it was taking coalitions to stop him. But in a reality France under his rule was in such a deep trouble economically that I would not take this type of a domination seriously.