Which document stated that Europe should stay out of American affairs, and that the United States would stay out of European affairs?

It is the Monroe Doctrine.

Explanation:

The key idea of the Monroe Doctrine is that the United States would see as a hostile act any intervention of European powers in the affairs of any nation of the American continent, it was something utterly unacceptable. At the same time, the US would not intervene in the affairs of Europe. These ideas were first introduced in a speech of president James Monroe before Congress in December 1823.