But if the Bible makes the case, doesn't that make racism a legitimate Judeo-Christian teaching?
No, that is you making a presupposition.
I follow that same Bible, but I have not seen anything saying you "must be racist" to anyone. I'm not racist, nobody in my church is racist, we have black and Hispanic church members in our congregation.
As John 7:24 puts it:
"Do not judge by appearances, but judge with right judgment.”
Whoops. I think I heard bellboy quoting a similar verse regarding homosexual marriage, but it also equally applies to racism as well. We are taught not to judge someone by their skin color, and thusly, anyone who tried to use The Bible as a way to justify racism were in fact doing the exact opposite of what it said.
Furthermore if you think this answer somehow justifies your case, reference the posts where I cited where the Bible explicitly defines marriage as between one man and one woman, then understand that our opposition isn't based on judging appearances or being bigoted, it is strict adherence to the Biblical definition of marriage.