That's debatable. There is, of course, the argument about the US getting sick and the world catching a cold. Yet I think the method of infection (to continue with the metaphor) is also significant. To be sure, as The National reported today, a decline in US GDP ain't exactly welcome anywhere. At the same time, problems in the US appear to affect us in three main ways: 1) the dollar peg, 2) foreign investment and 3) oil.
If the dollar gets weaker, so does the dirham (and the other Gulf currencies, with the partial exception of Kuwait's). But the dollar has remained strong through the financial crisis, and our buying power in, say, the UK or mainland Europe hasn't decreased. It has increased. So that's not really an issue.
If Americans make less stuff, they make less money, which means they have less free cash to spend. And that means, potentially, that fewer investors are willing to travel or put money abroad, including in the Gulf. Yet as far as I can tell, Americans haven't been a huge force on the Gulf property, tourism or investment scene, aside from their involvement in some of the major American mutual funds that invest out here: T. Rowe Price and Franklin Templeton, to name two. Americans also don't compose a big slice of the foreign property-ownership pie (according to government stats, Indians, Iranians, Pakistanis and Brits are the big players in Dubai property outside the Gulf). So I'd think transmission of losses through non-oil trade and investment flows wouldn't be a big issue, either.
The last thing, though, is oil. If Americans make less stuff, they don't need as much energy. Which reduces the demand for energy-producing substances, including oil. That, in turn, could lead to lower prices for oil and hence lower oil revenues for Gulf nations, which of course translates into lower GDP. That, it would seem, might be an issue.
Overall, that's not an overly negative state of affairs. GDP projections in the UAE are certainly lower than they have been in the past few years - there are some grumblings about the possibility of negative growth, as is projected by the IMF - but still a lot higher than in the US. Am I missing an important piece of the argument? Should we care about US GDP much more?