There is a technical sense in which they do use the metric system. All US units are defined in terms of metric ones, so that for example one inch is defined to be exactly 2.54 centimeters. However, this is not the same as using centimeters on a daily basis. Wikipedia has a nice article on the history of metric in the United States. There have been consistent efforts to convert the US to metric throughout the last 45 years or so. And, according to this source, even the founding fathers of Jefferson, Franklin, and Adams preferred metric.
The US metric study was a large study that ran from 1968 to 1971. Wikipedia says:
A 45-member advisory panel consulted and took testimony from hundreds of consumers, business organizations, labor groups, manufacturers, and state and local officials. The final report of the study concluded that the U.S. would eventually join the rest of the world in the use of the metric system of measurement. The study found that metric units were already implemented in many areas and that its use was increasing. The majority of study participants believed that conversion to the metric system was in the best interests of the U.S., particularly in view of the importance of foreign trade and the increasing influence of technology in the U.S.
This didn’t actually happen, though, apparently mostly because Americans were stubborn and mocked attempts to switch to metric. In 1975, the US passed the Metric Conversion Act, though it seems to have been weak legislation focused on federal agencies and industry. It led to the US Metric Board, which worked towards converting the US to metric for a few years before being abolished in 1982. Here’s Wikipedia’s picture of a speed limit sign from Florida in the 1980s:
It seems that road signs only last about ten years on average, so that converting them would not be prohibitively expensive, and Europe’s successful introduction of the Euro indicates that it’s entirely possible to teach people a new unit. The reason Americans haven’t switched over is simply that they don’t want to. If you ask them about it, they tend to give it less-than-sober-minded consideration. (You may be able to find an example without much difficulty.)
Efforts towards metrication in the US continue. The National Institute of Standards and Technology maintains a Metric Program. The US Metric Association (USMA) campaigns for metric.
When you’re telling your neighbour that you caught a fish that was “this big,” you can tell him about it in imperial or metric units just as easily. The main push towards metric in the US right now isn’t towards eliminating the inch or the pint from everyday use. It’s toward incorporating metric units into school curricula and international trade.
A recent “We the People” petition to make the Metric system the standard in the United States, instead of the Imperial system garnered almost 50,000 signatures. Patrick D. Gallagher, Under Secretary of Commerce for Standards and Technology and Director, National Institute of Standards and Technology replied in part:
Ultimately, the use of metric in this country is a choice and we would encourage Americans to continue to make the best choice for themselves and for the purpose at hand and to continue to learn how to move seamlessly between both systems.
In our voluntary system, it is the consumers who have the power to make this choice. So if you like, “speak” metric at home by setting your digital scales to kilograms and your thermometers to Celsius. Cook in metric with liters and grams and set your GPS to kilometers.
So it appears to be the official governmental position that metric units are superior, but realistically Americans’ dogmatic stance on the issue isn’t worth fighting, so using metric in everyday life is left as voluntary.