Immigration doesn't change the fact of natives dwindling away or not. A nation is the native people, not a name on a map. The exceptions are the United States and other new world ex-colonies. But there was a very dear price paid in blood and tears to make them.
America has had almost no assimilation of immigrants into the cultures which existed there before. Very good arguments have been made that the Enlightenment (or Age of Reason) did originate from native North American ideas and ideals being brought to Europe through cultural exchange, but it's not assimilation on the part of the settlers.