There was a reason no one wanted Arabia--it was worthless and not worth the trouble. It's been theorized Alexander wanted to conquer it before he died, but I can't imagine why.
It made sense for the Arabs to expand outward, to get ahold of some valuable land and great cities. It wouldn't make sense for anyone to go into Arabia proper, at least until the discovery of its natural resources.
Heck Oliver Stone practically had him prepared to conquer Britain and Japan at the end of his movie.