This question is complex, and that's magnified by the complexity of Arab identity.

In 1909, one Syrian was allowed to become a naturalized citizen because the judge in his case declared "I consider the Syrian as belonging to [the white race]... He is not particularly dark, and has none of the characteristics or appearances of the Mongolian race." In 1914, another immigrant from Syria was denied naturalization because the judge in that case declared "Syrians might be free white persons, [but] not that particular free white person to whom the act of congress had donated the privilege of citizenship" - basically, that Syrians were the wrong kind of white. (This particular judgment was reversed on appeal - after all, if you start letting everyone make their own calls on who's white then all kinds of mischief could result.)

That case, Dow v United States (1915), established that Syrians were officially white. However, this did not include Arabs from North Africa, or from the Arabian Peninsula, and the situation remained confused. Individual judges issued conflicting rulings.

The issue of Arab whiteness was resolved in 1943 when the Immigration and Naturalization Service declared finally that all Arabs were to be considered white - the first court test of this came in 1944, in the case Ex parte Mohriez. I'll include a bit of it, with citations omitted (you can click that link to read the full opinion). It's important to note that this ruling was issued just a few months after the repeal of the Chinese Exclusion Act:

In the understanding of the common man the Arab people belong to that division of the white race speaking the Semitic languages. Both the learned and the unlearned would compare the Arabs with the Jews towards whose naturalization every American Congress since the first has been avowedly sympathetic... The Arab people stand as one of the chief channels by which the traditions of white Europe, especially the ancient Greek traditions, have been carried into the present. It follows that even by the narrow criteria which were adopted in the opinions of Mr. Justice Sutherland the Arab passes muster as a white person.

...A person ought not to be encouraged to come here to live and to have children born here, who under the Fourteenth Amendment will automatically become citizens, unless we are prepared to give him the advantages of, and expect him to assume the obligations of, United States citizenship. It is contrary to our American creed to create a superior and inferior brand of permanent residents.

And finally it may not be out of place to say that, as is shown by our recent changes in the laws respecting persons of Chinese nationality and of the yellow race, we as a country have learned that policies of rigid exclusion are not only false to our professions of democratic liberalism but repugnant to our vital interests as a world power. In so far as the Nationality Act of 1940 is still open to interpretation, it is highly desirable that it should be interpreted so as to promote friendlier relations between the United States and other nations and so as to fulfill the promise that we shall treat all men as created equal.

For more, see Race and Arab Americans Before and After 9/11: From Invisible Citizens to Visible Subjects (Amaney Jamal and Nadine Naber, editors).