Wouldn't most Persians be considered to be white?
In the United States until recently I believe you're correct and people from central and southern Asia were regarded as "caucasian" due to language roots or something lame like that. Now I believe there are multple categories of Asian: East, South, etc. but it's been a long time since I took a census in America.
I use "Iran" and "Iranian" to refer to the country itself or things originating there, but "Persian" seems to be an acceptable demonym for the ethnicity. Persians can be found in places besides Iran, for that matter.