There aren't many real Caucasians in America.
Actually, "Caucasians" are from the Caucasas, which includes Georgia, Armenia, Azarbaijan and parts of southern Russia (i.e., Chechniya). You are right, Russians consider them non-white. (they refer to them as "black asses")
I would love to see Americans of Georgian, Armenian and Azerbaijani decent form a "Caucasian Club." What would the other minorities say?