The history of humankind is one long slog of the weak being conquered by he strong, or at least those with superior war-making technology subduing those with inferior defensive capabilities. This isn't a new development by Europeans in the 1600s, but describes the expansion of the Roman Empire, the spread of the Caliphate, the dominance of the Huns and Mongols, China's imperial footprint, not to mention thousands of smaller polities taking jumping on any advantage, no matter how insignificant to swallow up their neighbors. Even among the peoples that the Europeans colonized, wars of conquest, retribution or plunder were not unheard of, and in certain regions were a way of life. This is why I'm not comfortable with the term "stolen" when describing the European (and later American) nations taking over native lands. It's no more accurate than labelling the Saxons defeating the Britons, the Normans in turn defeating the Saxons, the Franks absorbing the Gauls and all the other displacements as stolen land, when it's just another in a long line of the powerful conquering the weaker groups.
The key difference is that in the fifteenth century the various European kingdoms looked at each other somewhat as peers. There were still wars of expansion, and the wars of religion were just around the corner, but a king who looked with lust upon his neighbor's natural resources still viewed the subjects of the neighboring kingdom as people. All of the various nations, kingdoms and empires were on par technologically, were all the same religion and were broadly similar culturally. Upon encountering people who were not as advanced technologically, and whom had never even heard of Christianity, the Europeans looked down on these nations, tribes and peoples in the Americas and Africa as savages, not even worthy of consideration. Europeans might steal produce from a native tribe, or kill someone, or rape a woman, or even encroach on land that belonged to the tribe, but would be absolutely flabbergasted that the natives would retaliate, and act as if they were the aggrieved party. Even with the millions of people already living in the Americas, there was undoubtedly still plenty of land on which the Europeans could settle, if they only respected the rights of those who were here first. But the Europeans wanted it all.
I'm not even sure that it was a racial animus as we understand the term today. The history of "whiteness" is long and complicated and calls for a whole 'nother blog post, but as a concept and identifier, no one would think of themselves as "white" for another 150-200 years. Europeans were certainly aware of people whose skin was darker than theirs, contact with south-of-Sahara Africa went back to Roman times, as well as with what we now Eurocentrically call the Middle East. Even within Europe the pale blond far northern Scandinavians and the darker Italians and Spaniards around the Mediterranean are opposite ends of a continuum of skin tone. No, the reason Europeans looked own upon the people they encountered in the Americas was their perceived level of civilization. To the Europeans they lacked any of the technological "advancements" that were common in Europe (such as gunpowder & steel swords), they didn't have cities (at least they didn't encounter any early on), their culture didn't seem to acknowledge individual ownership of land, and they weren't Christians...fair game!
Spreading Christianity was a major pretext for the European invasion. Conversion of the "heathens" to Christianity had been a goal of since the early days of Christianity in the Roman Empire. Now they had a whole new continent (two!) in which to "spread the Gospel".
Eventually, the concept of "whiteness" took hold. The Europeans were "White", the American natives were Red, the Africans were Black. They were "other", and "other" meant inferior. Of course they were viewed as inferior, no high technology, and more importantly, no Jesus. Why were they so technologically and spiritually inferior? They weren't white, and white equaled superior in their minds; everything else was inferior. Everything that followed: land grabs, forced conversions and assimilation, sequestration on reservations, breaking treaties - it all went back to the belief in the racial inferiority of the natives.
What if Columbus hadn't made it to the islands off North America? What would have changed? Probably little. Someone was going to eventually bump into the large landmass in between Europe and Asia. From all accounts Columbus was a horrible person, but he was a man of his time and very little would have changed if another captain had found America first. The kingdoms in Europe would have still been run by the same people who would look down on the inhabitants. Whiteness would have still emerged as a construct that resulted in White Supremacy and racism.
No matter. From the point of view of the descendants of the people that Columbus and other early colonizers encountered, the coming of Columbus was a horrible tragedy, the beginning of centuries of genocide and the wiping out of culture. Not a day to celebrate.