The people that were already here when the Europeans showed up certainly began to get pushed around before too long, and I don't want to make that sound unimportant; but the issue wasn't not having enough land for both the Indians and the Europeans, but rather who got what. As usual, the strong took what they wanted and the weak moved to other land.
Along with those white Europeans, of course, came people that didn't want to come; people who, even now, are still trying to make this a place that feels like home to them.
The real immigrations to America began in the 19th century. The railroads need for cheap labor brought Asians and Irish. Troubles in Ireland brought more Irish. Then came Italians and Germans and Scandinavians and Eastern Europeans, not to mention a steady trickle of English and Scots. By the turn of the 20th century, some people (who were already here, of course) got to thinking that maybe someone ought to stem the tide a bit.
Ellis Island finally was closed in 1954 and by then the population of the USA was over 150,000,000. Suddenly the vast amount of land didn't seem so inexhaustible anymore.
Today, the population is a bit more than twice that many, and God still isn't making any more land. What to do?
Proponents of open borders will quickly point out that early white settlers didn't have to worry about quotas and application forms. I agree. Life is unfair. First come, first served, so to speak. The reality is, the world and the USA simply aren't like they were back in the old days. The USA - and the world, for that matter - are getting filled up. If the people who are supposedly in power don't find a way to control the situation any better than the Indians did at first, then they will go down the same path as the Indians and lose the power.
There is a bigger problem, though. America's "secret" of how it rose to be a great world power is no secret at all: the driving force for making America great was its hungry immigrants looking for a better life, and willing to do whatever it took to advance from one generation to the next. What now? Immigration and diversity has always been the lifeblood of America's continual renewal. The people who advocate open borders have a point.