ok, let's define terms
when you use the term "race", what do you mean?
By race I mean a group of people defined primarily by skin tone, hair color and texture which may or may not include a precise country of origin, but usually doesn't.
Race doesn't have a biological definition, as there are no consistent characters within a race. Ethnicity and ancestral populations can be detected, but that's different from race.
White usually just refers to people with lighter skin tone, eyes (sometimes) and lighter hair, usually with origins somewhere in Europe. It used to be defined specifically as only northern Europe (specifically English and French ancestry primarly) and not eastern Europe (Poles and Irish weren't white originally).
As a race "middle eastern" would include most of the countries of the middle east, but probably more to the average white person since determinations are often based on a person's appearance. I would argue that this particular racial definition is also relatively new, many of these people might formerly have been considered white.