While our use of the word is similar, Europeans and Americans understand the concept of nationalism in fundamentally different ways. The American "nation" is rooted in identification with the state and the social contract. In Europe, "nation," by definition, is rooted in culture and ethnicity.
In Europe, state and nation are separate entities. For historical and legal reasons, in America that can never be the case. Moreover, in America nationality is a philosophical concept, rooted not in identity but rather in process and activity. Nevertheless, America does have an identity.
State and nation are used almost interchangeably in America. Almost. Specifically, the state is subject to the democratic will and the republican structure of the political process. Nationhood--or national identity--is an acceptance of the social contract that defines and shapes the process. American national identity therefore is embodied in the U.S. Constitution.
(Note: I am circumspect about the use of the word "will" in the third paragraph, but for now we'll table that question for another occasion, when we will also tackle the distinction between government and state.)