Ping for your opinion on the subject.
I think the question itself is too confused to be taken seriously. Is ethnicity the same as race? Is there a genuine scientific definition of race? How is ethnicity different from nationality or peoplehood?
Judaism (a word and concept that does not occur in ancient Hebrew, by the way) by most English language usages is religion. A convert, however, will have a very rough go without adopting 'ethnic' practices and outlooks. Critics of this approach almost always are expressing resentment of the fact that traditional Judaism insists on defining itself and its membership by its own rules without compromise.
The Torah itself uses the terms 'goy' (nation) and 'am' (people) to refer to Jews, as well as to Romans, Egyptians, Persians, etc.
One man's answer is at:
http://www.beingjewish.com/identity/race.html
Every so often I get this question. Usually, I am asked simply whether Judaism is an ethnicity or a race. One person phrased it to me this way:
"I am under the impression that "being Jewish" relates both to ethnicity (as in, someone is born Jewish), as well as a religion (as in, I converted to Judaism and therefore I am a Jew). Which is correct?"
What I cannot understand is: who cares? Why do people want to know? Of what relevance is this issue to being Jewish?
The Mesilas Yesharim, a seminal work in Jewish philosophy, has a chapter called "The Obligations of the Human in This World." He does not mention race or ethnicity even once in the entire chapter.
The fact is that as Jews, whether one is descended by maternal lineage from those who stood at Mount Sinai and accepted the Torah from Hashem, or whether one is a proper, full convert, or whether one is maternally descended from a proper, full convert, we are required to fulfill the Commandments of the Torah to the best of our abilities, and thus achieve self-actualization, holiness, and closeness to Hashem.
Words like "ethnicity," "religion," or even "race" are completely irrelevant to our obligations in this life. That's not the path to pursue. What is important is each individual's personal relationship with Hashem, and our relationship with Hashem as a People, as well as our relationships with our communities (roughly in that order)...