Xophone (Anglophone, Francophone, &c.) is used by people who require such things; but these terms are not in ordinary parlance. It appears that even the most anxious nativists don't use it: a search of U.S.English.org, which bills itself as "the nation's oldest and largest non-partisan citizens' action group dedicated to preserving the unifying role of the English language in the United States" doesn't yield a single hit on Anglophone.
We employ the adjective "English-speaking", and if pushed we'll call ourselves "English speakers"; but the fact is taken for granted rather than proclaimed as a point of pride. No doubt it's a sign of Anglophonic arrogance, but "being a native speaker of" English doesn't seem to need a label, at least to those who might bear it.