No, just like the language lawyering it's neither necessary nor sufficient, but the term is of course a bit fuzzy anyway.
I guess "xenophobic asshole" is a more accurate term but I thought "nazi" conveys enough meaning without resorting on language Americans might find offensive.
To clarify: there are valid arguments against immigration and there are especially valid criticisms on how immigration is handled (e.g. I would argue the German approach has for a long time largely consisted of hoping they either go away or magically assimilate on their own). But any argument that is based either on the requirement for immigrants to abandon their entire identity or on ethnicity alone is blatantly xenophobic and dehumanizing.
I guess "xenophobic asshole" is a more accurate term but I thought "nazi" conveys enough meaning without resorting on language Americans might find offensive.