I was trying to think of this, and for as long as I've watched football and remembered it (93-94), I can't think of any white RB that did this. My best guess would be John Riggins, but I couldn't find the answer on Google. Any one got any idea?
I can't find any archive, or list of 1,000 yard rushers in the NFL, much less one that calls out their race. But, I would guess it probably was Riggins, who ran for 1,200+ yards in 1984.