It is normally used in some set phrases like "Arabic numerals," but the official term to describe the nationality is Arab. Some people do use these interchangeably though (although some people may find it to be politically incorrect). Best to stick with Arab as a nationality and Arabic as a language :)
"He is Arabic" is definitely wrong but is used by some English-speaking people unfortunately.
At the very basic level, and as Alex beautifully explained it, an 'Arab' is a person (a noun and an adjective that relates to people/race), while 'Arabic' is a language and can be used as an adjective denoting an affiliation with the language.
So you can say "I am learning Arabic", "I love the Arabic language", "I don't like to watch films with Arabic subtitles", "There are no Arabic signs in this airport", "I have five Arabic textbooks" and so on.
You can also say "He is an Arabic-speaking person" but you cannot say "He is Arabic".
I apologize for bringing this topic up during this particular dicussion, but there does not seem to be a specific heading under which I can leave a comment.
It should be noted ...
In English, (depending on the particular country being discussed) there is sometimes no distinction between the 'word' for nationality or language, unless given context.
In English, the word Spanish can refer to both the nationality/country of origin and also the language.
"My friend is Spanish". = "Arkadaşım İspanyol".
"My friend is teaching me Spanish". = "Arkadaşim bana İspanyolca öğretiyor".
In one of the exercises, we are asked to translate the word Chinese from English to Turkish. If we give the answer "Çince" (language), our response is marked as being Correct, but if we answer "Çinli" (nationality) it is considered to be Incorrect.
Chinese can mean Çinli(nationality), as well as, Çince (language). Same goes for Spanish = İspanyol (nationality) Spanish = İspanyolca (language).
For these partıcular types of exercises, when translating from English to Turkish, both answers (whether referring to language or nationality) should be accepted.
In English it is impossible to distinguish what is being referred to (nationsaity or language) without context.
I have reported this problem, but the issue has still not been resolved. I have, therefore, posted these comments in the hopes that it may be helpful to others.
A question on the side...
Why do people get downgraded for posting (a genuine) question?! This often occurs in the discussion sections - I don't know why :0(
Asking questions is one (great) way to learn and no one should be 'punished' for doing so, or discouraged from doing so, as obvious as the answer may seem to others.
I am not sure what the word "Arabia" is in Turkish. If there is any such word in Turkish, then you can use it in your sentence instead of "Arablı".
The point here is, Arabia is a specific region (usually refers to the Arabian peninsula), so you can say that you are from that region. However, "Arab" is a race, not a place, so you cannot say "I am from Arab".