I dispute that avoiding obesity requires either exercise or high-quality nutrition. It is simply a matter of maintaining a caloric balance at a healthy weight. This can be effectively controlled entirely on the intake side. (See the nutritionist who lost weight and improved his bloodwork on a gas station junk food diet. [0]) Other countries with varied labor and economic conditions (presumably among them are some you would not label medieval) are experiencing the same obesity pandemic. There is nothing unique about the US in this regard other than it has been leading the trend.

[0]: http://www.cnn.com/2010/HEALTH/11/08/twinkie.diet.professor/