Excellent question.

The Vanguard 500 index fund buys each stock in proportion to its market capitalization, which gives you optimal diversification among these stocks. To the best of my knowledge there is no easy-to-follow mathematical formula for diversification across countries especially given exchange rate risks. Also, you probably shouldn't use passive investment vehicles (like index funds) for investing in corruption-ridden financial markets such as China's.

But you probably should invest internationally and use the index approach when investing in stocks in rich countries. Although given the international nature of the 500 biggest U.S. companies (which the Vanguard 500 index fund consists of) you do get a lot of international diversification when you invest in it. If you are a relatively small U.S. investor it might not be worth it to invest in non-U.S. stocks, but I'm not sure about this.

Vanguard provides VGTSX (Total International), which attempts to invest in non-US markets in a balanced and diversified manner. I consider an international fund crucial to maintaining diversification, though many people feel it's too volatile. It was the only thing I found in all my reading where people had widely varying opinions with no true consensus, in which case you should go with your risk tolerance.