The discovery of the new world was an accident by Columbus looking to find a direct route to the goodies in Asia. The Chinese and Indians produced nice things and had the climate for spices. As these goods were traded west each merchant added his mark-up, and by the time they got to Iberia (Portugal, Castile, Aragon, Navara (before 1492)) they were very expensive. The Portuguese started trying to go around Africa so they could profit by undercutting the Italians and Greeks and Arabs.

In 1453 the Turks took the last of the Roman Empire when Constantinople fell. This made trade through the old roads harder and thus more expensive.

Another reason for exploring the coast of Africa was gold. The richest man in Human history was Mansa Musa who was the king of Mali in the 14th century. His legendary wealth was so vast that when we went on the Haj, according to legend, he caused inflation in the price of gold in Cairo by flooding the city when he stayed there on his journey.

Because of the gold mines funneled their wealth through his kingdom and his kingdom was so efficient the legend of the wealth of Africa spread to Europe, which was the poorest part of the world at the time. Naturally coastal powers would go looking for this gold.

Let us say by some miracle that the Romans keep the Turks from taking over. The development of European shipping and trade voyages would have pushed some one to try to go west. While most people in the know knew the approximate diameter of the earth and thus the distance to china, there were enough people like Columbus who believed the smaller estimations that were spreading around. Also we have to remember that the nordics got to North America in longboats in the peak of a little ice age so perhaps some one in northern Europe would go looking for this place and find north america. (A fun what-if would be what if european first contact with the main-land was in what would become newfoundland rather than mexico?)

It was only a matter of when north america was discovered, not If.