The short answer to this question is no. By "post-American world," Zakaria describes the "rise of the rest," other nations that will become economic powers alongside the United States. While this has led to a great deal of anxiety in the United States, Zakaria emphasizes that it does not mean that the United States will not remain powerful, important, and wealthy.
The "rise of the rest," in short, does not entail the inevitable decline of the United States. Rather, Zakaria views the rise of other nations as an opportunity for the United States. But it will take a sort of paradigm shift on the part of the United States, and Americans, if this is to occur. Americans, in short, have to be willing to abandon their insular outlook and their view, long reinforced by politicians, that the global economy is a zero-sum game.
Despite the growth of economies in China and India, the United States will remain an economic power for the foreseeable future, and it is especially strong in developing innovations, especially technology, that can be marketed to emerging economies. So, Zakaria does not think that the issues posed by the "rise of the rest" are insurmountable, but he emphasizes that the United States must embrace free trade, relatively open immigration policies, and other aspects of a globalized world that will allow it to continue to flourish.