Race is a fraught concept that has no basis in objective reality. Social scientists usually use the term "socially constructed" to describe the idea behind race to mean that it's developed based on the society in which the idea takes hold. This is why you have such varying ideas of race (e.g. Nazis thought Europeans were divided into many races, and that Jews were an entirely different race, but in the United States, all Europeans are generally considered to be white and Jews, at least Ashkenazis, specifically are also considered to be white, and that is the extent to racial divisions among Europeans - that is to say there aren't any). So you're not going to find a "definitive" answer to this question regarding Arabs. It will depend on who you ask. Most Arabs tend to ascribe to the "white" grouping in the United States, but I have no idea how they might describe themselves in other countries. Obviously, the Afro-Arab and Asian-Arab populations might feel differently. The question sounds like you're asking for yourself, so I would suggest you could answer whatever you feel like you are, because that's really about as objective an answer as you're going to find. You could also mark "other" if that would make you feel better.

To go a bit further, one's race is determined by any number of socially constructed ideas. An example I often use with my students is the question of "what makes a person black?" There is no good, objective answer to this question. For much of US history, even a single African ancestor would mark a person as black. But why is this? A single European ancestor does not mark a person as white, so why should it apply to black? Further, it's not entirely culturally based either (i.e. the culture of the parents). Barack Obama is a great example here, as he is the son of a Kenyan man and a white American from Kansas. He was raised by his white mother and white grandparents, yet he identifies as black. Now, of course, part of this is because the racist notion that any blackness makes someone black (i.e. I mean that this "choice" wasn't completely his, but was influenced by the way others would see him), but culturally, he would have been just as correct to claim he was white, given his upbringing and ancestry (all other things being equal, that is), or even African (as opposed to African-American) because of his father's status.

I agree with /u/Jan_van_Bergen on the social constructedness of the "Arab race" or "Arab ethnicity", as with all races and ethnicities, but we can also consider how that social construct exists for and among Arabs themselves.

Minimally defined, Arab is nearly synonymous with "Bedouin" that is, a desert-dwelling, Arabic-speaking tribal nomad of the Arabian peninsula and the deserts that extend into the Levant and parts of North Africa, or their descendants. The word "Arab" itself referring to the desert, etymologically. A good example of that minimal definition being the fact that until the 20th century it would have been unusual for most urban Egyptians, to give one example, to define themselves as "Arabs." They would have been Arabic-speaking Muslim Egyptians, yes, but "Arabs" lived out in the desert.

Maximally defined, however, "who speaks Arabic is an Arab." I've also seen that line attributed to being a hadith but never come across it. If it is I can't imagine it's sahih (sound). This is the Arab-ness of many versions of pan-Arab nationalism and of the kind that unites the Arab League. It's how you end up with the former Lebanese PM Tammam Salam being just as Arab as, say, Sultan Qaboos of Oman.

Day-to-day perceptions, in recent history, are all over the place, but somewhere in the middle. Within the Arab world, as in the "White European World", there are just as many national and quasi-ethnic racialisms as you would expect for a region with a couple dozen countries, hundreds of millions of people and any number of historical grievances and rivalries. So Egypt might be the heart of pan-Arab nationalism, but it's still common to hear that all of the other Arab states are just tribes with flags. Sayyid Qutb who was an important Islamist intellectual and an Egyptian from Upper Egypt (i.e. southern Egypt, which is traditionally darker skinned), could simultaneously maintain the cognitive dissonance to hate the United States for segregation because it was unfair racially, hate segregation specifically because he thought that as an Egyptian he should not ethnically be subjected to it because Egyptians weren't black, and hold incredibly racist views about African-Americans.

Then there are the many (but often Christian or other religious minority) groups who hold that they are in fact not Arab at all but actually Phoenician or Assyrian or whatever.

So yes, to circle back to /u/Jan_van_Bergen, it's ultimately a social construct and an identity, but one that is interesting socially and historically.