Not really! That's the conclusion you'd have to draw from a new Gallup poll today (via Economix). Take a look at these graphs and tell me if you see any kind of relationship between Dow fluctuations and presidential approval ratings. I certainly don't.
Catherine Rampell wonders whether this flies in the face of the whole Economist-in-Chief Theory that voters re-elect their leaders when the economy is doing well, and vote them out when it's doing poorly. Here's another thought. The stock market is often considered a leading indicator of economic conditions. Unemployment, which probably weighs more on voters' minds, is a lagging indicator. (You can compare S&P fluctuations to unemployment over the last few decades here.) So if stock market movements anticipate economic health, which in turn guides unemployment, which in turn colors voters' opinions of their leadership, it makes sense that each tick in the market shouldn't necessarily hold hands with presidential approval ratings.
Finally there are more specific explanations for each president's approval trajectory. Despite a great run for the stock market in the 2000s, W. also had Katrina and Iraq dragging down his approval, especially in term two. That dip in Reagan's numbers in early 1987 is right around the time when Iran-Contra was dominating headlines. The Iran hostage crisis obviously devastated Carter. We might vote with all eyes on the economy, but we approve with our eyes on the day's news.