Actually, Greek mathematicians had already proved the Earth was round long before the medieval era. They had noticed that a pole placed vertically in the ground cast a shadow of different lengths in cities far enough part. By performing precise measurements in these cities and combining their data, they even calculated the size of our planet using trigonometry. They weren't spot-on, but the inaccuracy was surprisingly small for such a simple technique. In fact, the famous Archimedes paper The Sand Reckoner, which tried to estimate the number of grains of sand on every beach on the planet, was only off by about 100% -- not bad when you consider that he had no idea how many continents there were nor the shape of their coastlines!
The myth that all European explorers were working with the belief in a flat Earth is also nonsense. Like Archimedes, European explorers had no idea how many continents there were or how they were shaped, and so were seeking routes to India and any other potential trading partners they might find. Anyone who sailed knew that a ship's body (and then mast) slowly vanished over the horizon, which meant the world must be curved.
I recommend consulting the magazine The Skeptical Inquirer, which only recently published an article on the modern belief in the flat Earth and its origins.