It has, in some very important ways, but this is a question whose socio-cultural, religious, and political aspects very much overshadow straightforward linguistic analysis. You can also see the opposite situation with Hindi and Urdu: they're totally mutually intelligible, but for the same kinds of reasons are considered distinct languages by both communities.

From a linguistic standpoint, the Maghrebi (north African) and Mashriqi (Levantine) dialects of Arabic are about as mutually incomprehensible as Spanish and Italian. There are two dozen or so Arabic dialects in current use, roughly divided into seven macro-groups (Maghrebi, Sudanese, Egyptian, Peninsular, Levantine, Mesopotamian, and Andalusian); Wikipedia has a nice chart demonstrating dialectical differences in a couple of simple sentences.

Speakers across the Arab world code-switch between their native regional "dialect" and Modern Standard Arabic, which is itself descended from the Classical Arabic of the Quran. Even MSA has some regional variation in how it's pronounced, although my understanding is that that's more analogous to differences in accents amongst American English speakers than anything else. CA and MSA are also used by millions of non-Arab Muslims around the world (e.g., Berbers, Kurds, Malaysians, Albanians, Indonesians, etc.) for primarily religious reasons, but who otherwise communicate in their native non-Arabic languages.

There's lots of good stuff on code-switching, diglossia, and modern language shifts in Arabic in the essay collection Language Contact and Language Conflict in Arabic, ed. Aleya Rouchdy (2013). It's available in print and in PDF on Google Books.

EDIT: I forgot to mention that the first signs of divergence in the Arabic dialects began appearing ~1400-1500 years ago, roughly coinciding with the first wave of Muslim expansion in the early Middle Ages. For comparison, the Romance languages began diverging ~2000 years ago.

One addition I would make to /u/suddenly_mozzarella's post, which I could comment on some points but I think is basically on the money is that nobody writes in Arabic dialect. Even the Juba Arabic dialect (of South Sudan), which I think might be one of the oddest sounding versions I'm Arabic that I've heard, its local language publications are put out in Modern Standard.

I'd also note that the use of "diglossic" to describe Arabic is sometimes disputed. The classical "diglossic" language is Greek, which has both an ancient and modern form. This was shoehorned into an Arabic context that is quite different. I think multi-glossic or language continuum might be a better description. Particularly because outside of a media or religious context it would be very unusual to "code switch" from colloquial to MSA. Rather, you have gradations within each dialect between, say, street slang and how you'd hear a well-educated lawyer speak.

Arabic media also plays a binding role. Over the course of the 20th century the center of Arabic media has shifted at various times/extents between Egypt, Lebanon, Syria, and the Gulf. An Arab child in Iraq watching the Lion King will have grown up hearing the Egyptian dubbing. Not to break 20-year rule stuff, but as an example, the dubbing of Frozen was done in a far more stilted MSA. There's a good essay on that issue here. But my point being that having an understanding the most common regional dialects is the default, even if each individual wouldn't "code switch" into a different dialect entirely.

Follow up to /u/suddenly_mozzarella 's excellent post:

The Roman Empire is much, much older than the Islamic empire. The main bulk of Islamic conquest occurred some 1200 years ago, while the Roman Empire was at its peak 2000 odd years ago. There has been much more time for the former Roman provinces to develop their own cultural and linguistic identities than there has been for the former Muslim ones.

You might want to ask in /r/asklinguists, too.