In summary, he could be one of the following:
1.Persian. (that’s what Wikipedia & many other sources say, supported by the name of “Majousi” in his full name, although that’s highly controversial & by the fact that Khwarizim was part of greater Khorasan in the early days of Muslim Caliphate).
2.Arab. (that’s what Encyclopedia Britanica, Microsoft Encyclopedia and many other sources suggest, supported by the fact he never spoke or written a word in Persian).
3.Neither Arab nor Persian, but a native Transoxanian Khwarizmian (supported by the fact that this is he is name & the obscurity of his origin, i.e he would have bragged about his racial origins if he was Arab or Persian because both of these ethnic groups were important & influential at the time of Caliph Ma’amoon, who himself was half Arab & half Persian).

An additional source of confusion is failure to understand that the Arabs back then, just like all colonial powers in history, were colonizing all of their conquered territories and Transoxiana was no exception. Take Abdulqadir Al-Gailani for example, he was born in Gailan in western Iran but ethnically he was Arab. He never spoke or wrote a word in the Persian language (although it was the major language east of the Tigris because the Arab status had declined compared to the early days of Muslim Empire). Wikipedia mentions him as a Persian but fails to mention that ethnically he is Arab.

Lastly, he was linguistically Arab (as he did not speak or write any of his works in Persian & lived amongst Arabs for most or all of his life), so even if he was ethnically Persian or Transoxanian or any other nationality, he is an Arabized Persian/Transoxianian. Persian nationalists won’t like this idea, but it seems he was fine about it, because if he wasn’t, meaning that he felt forced to write in Arabic for any reason, he would have at least wrote or made a reference of his Persian ancestry. He didn’t do that, however, simply because he didn’t feel connected with his Persian origin or because he didn’t have this origin in the 1st place.
That, I think, gives the Arabs a slightly stronger right of celebrating the legacy of that man, who is probably the greatest single scientific figure of the Islamic era & all of us who relate to Islamic culture, should be proud of.

regards Tommy B