In reality, it was in large part an accident of geography. Europe is essentially a mountain-ringed peninsula, and Western Europe was largely protected from the migrations and invasions from the Asian Steppe that did so much damage to Medieval Islamic civilization, China, India and Byzantium. The latter was critical because a lot of learned people fled to Italy as the Eastern Empire collapsed, and most historians feel there is at least some connection between the Renaissance and the collapse of that empire. But the damage done to Medieval Islam was incalculable; great centers of learning like Baghdad, with their libraries and scholars was so great that even the height of the Ottoman Empire (under the descendants of Turkic invaders) may have acquired a lot of territory but never really recovered the advantages that the high Muslim era had produced.
Why all these contortions were going on, Western Europe was largely left alone. It had its crises to deal with, like the Black Plague, but in an ironic way, the plague was one of those once-in-a-millennium demographic makeovers that remade Europe. While the rest Eurasia, which had been, up to the late Medieval period, as advanced and in many ways much more advanced (believe me, you would rather have had a serious illness in medieval Madrid or Baghdad than Paris or London). But the fundamental tools of Western domination were all invented elsewhere (hence Diamond's "Guns" and "Steel" in "Guns, Germs and Steel").
Probably the single biggest factor in Europe's dominance wasn't Spanish, French or English weapons and ships, but that greatest of Renaissance innovations; the modern bank, invented in Italy. The creation of the first banking and capital markets was a revolution that allowed the infant European empires to fund colonization. I will give Europe that, but everything else; the gunpowder, the steel, the grain crops, the livestock, heck even the writing systems, were all invented outside of Europe.
At any rate, I digress. The fact is that Europe got bloody lucky through a sheer accident of geography. The hoards of invaders coming out of the Asian Steppe did manage to smash the periphery; such as Russia under that Tatars, but Western Christendom was left free to build up more advanced economic systems, and it was the mercantilism that was transformed into early capitalism that was responsible for much of Europe's dominance.
I won't deny that Islamic civilization partially collapsed; though in its defense, as I mention above, that collapse had a lot to do with external factors. Even Islamic Spain (the most advanced state in Western Europe at the time) largely failed because the degradation of the rest of Islamic civilization left it vulnerable to the Reconquista. And one only has to look at the delightful and enlightened replacement the rule of Ferdinand and Isabella was to Moorish Spain to see how far much of Europe had to go.