In fiction you could have Medieval era lasting for million years if you wish. In reality surviving and expanded Roman Empire gives better chance for stagnation than OTL Medieval Europe.
This is inaccurate as the Roman Empire was around during the Middle Ages. This was what people call the Byzantine Empire which was just the Eastern Roman Empire that still lived on in the Balkans, Anatolia, and Southern Italy. It fell in 1453 while its splinter states the Empire of Trebizond and the Principality of Theodoro fell in 1463 and 1475 respectively. The Eastern Roman Empire was the most advanced state in Christendom for most of the Middle Ages as well (Up until 1204). The central government and the core structures of the Imperial Army survived the Migration Age unlike what happened in Western Europe. These old systems from the Classical Era were adapted to suit the present needs of the Romans which was why they were switching from heavy infantry to cavalry based armies.
Even in the Classical Roman Age, technology was still advancing as people discovered new things about their environment, or came up with new military designs. Humans from 2,000 years ago are just as capable and intelligent as the humans of the 21st century. The only difference between us and them is that we have the collective knowledge of the millennia of human development on this planet. The Romans if they were transported to our world, would still be able to understand things in modern architecture as we use principles developed during the Greco-Roman period. Its simply impossible to simply stagnate all human development like that unless there were some kind of demographic collapse due to a mass pandemic or natural disaster on the scale of say a supervolcano erupting, or an Asteroid hitting the Earth.
There is. Ottoman Empire was built up in 14th century, pre-gunpowder. As late as the 1422 siege of Constantinople, guns were a minor factor.
Without gunpowder, cracking Constantinople takes more persistence in blockading, building up Osman navy and mopping up ports to serve for resupply, but it would be done.
The Ottomans were determined to take Constantinople no matter what. In the 15th century, the city was a depopulated wasteland and it wasn't even a major trade center as the large bulk of Black sea trade was routed through the Genoese controlled straits of Gallipoli.
Even if the Ottomans were to arise, the Italian City states dominated the trade in the Mediterranean which increased resentment in other European merchants. The Pressure was already there to look for new trade routes that other European states like Portugal could use without the Italian middleman. If the Roman Empire survived it too might eventually look for other trade routes as well. They had a huge trade deficit with the Persians who grew rich as the bridge between China and the Silk Road Trade. This was why Egypt was important. It provided a direct sea route to the Indian Ocean trade. The Romans might go West if the Persians or some other Eastern Power is around that cuts them off from the Silk Road Trade.
