The U.S. Dollar & Stock Markets
It should be noted that as far as the U.S. stock market is concerned, this causation goes both ways. The main reason for this causal connection is that investors treat the U.S. dollar as a "negative beta" asset-one that should fall in value whenever stock markets rise in value and vice versa. Investors reduce their demand for U.S. dollars whenever stock markets rally-and they increase their demand for U.S. dollars whenever stock markets sell off. In part this reaction reflects something of a self-fulfilling prophecy: Investors sell dollars in response to stock market rallies because they believe it should do so, causing it to happen. Another factor is the theory that I described here with regard to the yen (the other main "negative beta"-currency) and stock markets and that also applies to the U.S. dollar.
However, it is not only the case that stock market movements causes the U.S. dollar to move in the other direction, it is also the case that dollar movements causes at least the U.S. stock market to go in the other direction. While not all U.S. companies benefit from a weaker dollar, most do. Either because they export goods or services from America and/or because they have foreign subsidiaries whose earnings in dollar terms increase whenever the dollar weakens. In both cases (but particularly in the former) this means that their profits in nominal U.S. dollar terms increases whenever the dollar weakens-and that they decrease when the dollar strengthens.
For other stock markets however (Except those in countries with currencies pegged to the U.S. dollar of course), movements in the U.S. dollar will have a positive causal effect (meaning that it will cause them to move in the same direction, not necessarily go up as a weaker dollar lower profits for most companies outside the U.S.), somewhat counteracting empirically the similar negative causal effect that stock market movements have on foreign stock markets.
Because most investors care more about market momentum than about underlying fundamentals, the causal link is probably stronger in the short term in terms of the effect of stock markets on the dollar than in terms of the effect of the dollar on the U.S. stock markets. However, in the medium long term, any "permanent" shift in the U.S. dollar's (real) exchange rate will result in U.S. stock prices moving in the opposite direction, while causing other stock markets to move in the same direction.
The negative causal connection between the strength of a currency and the local currency value of the stock market really applies to all countries. By contrast, the negative causal connection between the stock market and the value of the currency only applies to the U.S. dollar (and currencies pegged to it) and the yen and to a lesser extent the Swiss franc. For many other currencies, particularly the dollars of Australia and New Zealand, a stronger stock market will have a positive causal effect.