Noun: A person born in a specified place or associated with a place by birth, whether subsequently resident there or not.
Noun: A native or citizen of the United States.
Adjective: Of, relating to, or characteristic of the United States or its inhabitants.
So, a "Native American monk" is simply one who was born in the United States, no?
Ajahn Thanissaro, and as I recall all but one or two of his monastic students are American by birth.
So, that's a start.
Metta and Anjali,