Yes, let's agree with George Washington on this point.
Washington made the statement you quoted as part of his farewell address to the nation, delivered September 17, 1796. The idea in his statement was that whether American by birth or by choice, we rally around the idea of America -- that we are working together as one for the common goals and ideals that America has. The United States of America has always been that kind of nation. The unity of the United States comes from the ideas that together we hold dear, rather than from all persons in the country having the same ethnicity or cultural heritage from which they originated.