Except for Native Americans, would American literature be considered an immigrant narrative? Ralph Waldo Emerson was born in the United States, would his "Self-Reliance" be an immigrant narrative?

You could consider Ralph Waldo Emerson's "Self-Reliance" to be part of an immigrant narrative, depending on your interpretation of what this term means, and indeed what an immigrant is. In modern American literature, an "immigrant narrative" usually refers to works produced by people who have entered a dominant culture, that of the United States, as part of a minority. Threads common to immigrant narratives include the theme of trying to assimilate culturally; the difficulties of trying to strike a balance between one's native or ancestral culture and American culture; and the anxiety, particularly in second-generation immigrants, of feeling torn between two worlds. Certainly, being born in America does not invalidate a piece of work as an immigrant narrative: second-generation and third-generation immigrants go through experiences which represent different parts of the immigrant narrative to that of their forebears, who actually journeyed to America from elsewhere. In this sense, the fact that Emerson was born in America does not mean that his writings cannot be "immigrant narrative," but, on the other hand, we can see that they do not convey the same themes and concerns as most immigrant narratives today.
Immigrant narratives, strictly speaking, revolve around leaving one place, traveling to another, the culture shock of first exposure to a new culture, and the subsequent confusing of identity. The final stage of many immigrant narratives is the rediscovery, and increased self-awareness, of that identity.
The early Americans, unlike modern immigrants to the US, were not necessarily examples of minority groups entering a disparate dominant culture. On the contrary, several groups brought their own culture and blended these, transforming the blend into a new culture that became the foundation of the new American nation.
I think there are reasons to argue that certain core pieces of American literature are indeed part of a unique immigrant narrative and do share some themes with true immigrant narratives, owing to America's founding by immigrants.
The story of a new culture, and national character, developing is not the same as that of, say, a Filipino immigrant struggling to assimilate into a dominant American culture, but it is an important narrative of its own about the birth of a new country and the ways in which immigrant stories can combine to become something new. In "Self-Reliance," Emerson expounds upon some of the values that had developed in the US since its inception, particularly those of "individualism, personal responsibility, and nonconformity." Individualism, in particular, is now a core tenet of American culture, perhaps the reason American capitalism is so entrenched in and certainly connected to the American Dream: the idea that any person can make whatever he wants of one's self, provided that he or she is willing to work hard. In "Self-Reliance," then, we can see an early stage of American identity being forged, which could arguably have roots in the legacy of the first immigrants to America. Similarly, this character mixes with other immigrants encountering it later on; thus becoming part of any immigrant narrative in which, for example, a Filipino in America becomes a Filipino-American, no longer half one thing and half another, but part of a synthesized culture that is unique in itself.
In summary, then: no, all American literature, particularly modern literature, is not exactly part of an immigrant narrative. Immigrant narratives are understood to share certain themes and elements which writing from the dominant culture does not share. In Emerson's "Self-Reliance," however, we can see some of the themes of modern immigrant narrative presented in a different, interesting, and very relevant context. So, while "Self-Reliance" would not be classified as an immigrant narrative according to the general understanding thereof, it could be considered part of a different kind of immigrant narrative dealing with legacy and the interrelation of identities—and is certainly worth studying in that context.
http://coursesite.uhcl.edu/HSH/Whitec/terms/I/immigrant.htm