[jcranmer]

English grammar generally follows the rule that proper nouns and derivations thereof are capitalized. So the proper way to write it would be white, black, Asian, Hispanic (also Latino/Latina), Native American (or Indian or Amerindian), Pacific Islander, etc. Note the difference between native American and Native American.

[DrScump]

  proper nouns and derivations thereof are capitalized

Not when used as adjectives, as in the given example. Had it been hyphenated, creating a lone noun, "Asian-American" would work.

[dragonwriter]

Proper adjectives (adjectives derived from proper nouns) are generally capitalized in English.