There are so many problems with this one, I hardly know where to start. Obviously, if you were born in this country, you're not an immigrant, you're a native American. It cannot be said that non-Indian people who were born here are not native Americans. It can't even be said that the Indians originated here, since the origin story we have for them is that the Indians themeselves came over here originally from Asia. And Indians who were born in some other country are not really native Americans as far as being natives of this country are concerned, they are only members of the same race that some people call "Native Americans", what I would call American Indians. As I am part Indian myself, I have no real reason to worry about what somebody else might think of that.
Finally, the statement quoted above comes from someone who so far has failed to prove that he himself is a native American, something that is supposed to be neccessary in order for him to hold his own office.