Everything is connected somehow. If things are bad in the stock markets in other countries, it will be noticed. America is the best country in the world but it can't survive by itself. It needs a strong economy and when it doesn't have one, it needs help from other countries that are doing well to help it out.
Yes, international stock markets impact American stock market activity. In today's connected world, many American companies work with international companies, and a failure in an international stock market will also affect American companies, thus causing the American stock market to decline. This can also occur in a positive way, where success in international stock markets will cause American companies to prosper, causing a boom in the American stock market.
Yes, international stock markets have an impact on the American stock market. Just watch the stock market indexes. When the international market goes up or down, often times the American market follows. Because the many of the international markets are ahead of the American market time-wise, investors closely watch what the international markets do before the American markets open, and they are impacted accordingly.
When it comes to the activities of the American stock markets, one variable that plays into the thinking of the brokers and the investors is the international stock markets. Keeping an eye on what the rest of the world is doing with their money, people who work and invest in the stock markets want to watch for new trends or issues on a global level.