"Nor can it be doubted that it is the inherent right of every independent nation to determine for itself, and according to its own constitution and laws, what classes of persons shall be entitled to its citizenship."<quoted text>
The US Constitution has never recognized dual citizenship. That means that foreign citizenship has no effect on US law AT ALL. This does not strip someone of their foreign citizenship. It just means that people who are dual citizens are just as much US citizens as people who are only US citizens.
Looks like foreign citizenship does have an effect on US law. The US Government can not strip a citizenship from anyone unless requested by that person.