Ethnocentrism shaped European encounters with native people (such as the Native Americans) by giving the Europeans the sense that they could treat these people as they wanted. They felt superior towards the native peoples and treated them more like objects than like human beings who were equal to them.
This can be seen quite clearly in the example of the Americas. Native peoples in the Americas were typically either pushed out of the way (or simply killed) or were assimilated into European-dominated societies, but at the lowest rungs.
In what is now the United States, ethnocentrism led to the idea that the natives were not legitimate owners of the land and could be pushed aside as the Europeans wanted. This often involved wars to kill natives as well as to push the survivors off a given area of land.
In what is now Latin America, natives were used as more or less captive labor. The system of encomienda gave Spanish landowners the right to the labor of a given population of natives. This was supported by the idea that the natives were inferior to the Spanish and therefore could be disposed of as the Spanish wished.
In these ways, ethnocentrism made Europeans treat native peoples like objects. The Europeans felt superior and so they did whatever they wanted to do (whatever was most convenient for them) to the natives.