I was just thinking about it and couldn't come up with an answer. My Asian history knowledge is basically non-existent so I'm curious.
So in the very beginning (3000 BC) the cultural and main center of the world was in Mesopotamia and Egypt etc. so the big antique civilizations in the Middle-East.
With a big leap forward came the Greeks then the Romans shifting this center closer and closer to the heart of Europe.
Then when the Roman Empire fell the Arabs preserved the legacy of the antique world, then when the Dark Age passed for Europe they could newly discover the knowledge thanks to the Arabians while the Arabs began to dwindle in relevancy until the Ottomans.
And after that there's only Europe... discovering America and all.
I know that before Christ the Asian cultures were pretty big I mean India and China. And they had pretty strong inventions at hand, like gunpowder for the Chinese. And their religion was by no means as restrictive as the Christianity during the inquisition.
So how come that Asians didn't discover America ? (they are even closer to it than Europeans)
Is the relevancy of Asia so small because everything in school is taught with a "European-eye"? (as in views centered around Europe). I come to think of it Asian art was never taught in school. Is it because the culture is much less refined?
It most likely is, like you say, that history in schools is taught with a european eye, in history this is called the eurocentric view. Even the epochs we have are constructed mostly around european history and often unrelateable to conditions in other parts of the world that western history tends to turn a blind eye to. There is however a new field of research that is called global history which attempts to create global historical framework and aims to show the global consequences of certain local historical events.
I'll address just the question about why the Chinese did not discover America. There is some debate about whether they actually did, but they indisputably did not colonize those continents.
Europeans went to Asia because Asia had goods that Europeans wanted. The Ottomans taking Constantinople in 1453 created a heavy disruption in European<->Asian trade routes, traditionally dominated by the Venetians. This event created an opportunity for the Portuguese and Spanish to seek out new ways of getting goods from Asia.
So the New World was discovered because Asia had goods that the West wanted, and Asia was less excited about European goods. The New World just happened to be in between one of the routes from Europe to Asia.
It perhaps seems shortsighted at the time, but Asia seemed content to have the Europeans come to them. This spared them the cost and risks of international trade. Ultimately this inspired Europe to improve their shipbuilding such that they had navies that could dominate the world's oceans, but that was hardly obvious at the time.