Terrorism? Blame Alexander

Alexander the Great never conquered the Arabian Peninsula. He died of illness in 323 BC before he had gotten around to executing a campaign against Arabia.
But suppose he had lived a little longer, enough to invade and, in all likelihood, conquer Arabia? Let's do a quick alternate-history inventory of the possible long-term consequences:

• An at least partially Hellenized Arabia brought into the orbit of the Mediterranean world
• Arabia as a province of Egypt or another Alexandrian sub-state after Alexander's death
• Roman control of, or influence over, Arabia as part of the Empire or a pacified ally
• The introduction of Christianity into Arabia through the Roman Empire, filling the monotheistic vacuum among the Bedouins
• No rise of Islam
• No clash between Christian and Islamic civilizations
• No 21st century Islamic grievances, or Islamist terrorism.

I admit that this is a stretch, but that's the fun of alternate history. Of course, I haven't sketched all the possible consequences of this change -- there's also a plausible "no Islam = no preservation of Greco-Roman learning = no European Renaissance" chain of reasoning that should discourage those of you with time machines from acting precipitously.