Has online marketing caught up with traditional marketing in the Middle East?

Excellent question! Contrary to popular belief, online marketing does not differ too much from traditional marketing.
Whether online or traditional, marketing has always been about user experience, customer engagement, and encouraging sale through a substantive value proposition.
The tools have changed, but the principles are still there.
It is therefore hard to say that traditional marketing is completely dead.
As any other business practice, marketing is meant to change.
You should rather think of it as an evolution: marketing finally caught up to where customers are.
The two types of marketing should be working hand-in-hand to create better relationships with customers, and truly add value.

In terms of absolute dollar value spent, non-online marketing still commands a far greater share of pocket in the region.
Estimates vary (a quick google search will show), but online marketing in the region is no greater than 10% of total advertising spend.

Online marketing has not yet caught up with traditional marketing, but it is going to, as a growing numbers in the Middle east are now a ware of it's significance especially after the so- called "Arab Spring"

The question's very relevant, but 'caught-up' - in terms of % of marketing budget allocated to Online no way! And I feel its some way off still.
IMO here's why - marketing's end objective is to drive sales or increase preference and affinity shares - while that is a function of brand equity, product fit, credibility scores and promotional relevance, importantly a medium works primarily if end-customers' buying behavior and attitudes are aligned to the medium's strengths/ evolution to impact decision making in that region.
In this region (and this is solely my view) the retail experience is still a very big thing.
Consumers largely, still like to touch-feel a product or service before making the purchase decision.
And, its not just that, stuff that impacts decision making like unbiased user reviews, opinions from friends are still lacking in the online platforms here, while that's fast changing, there will be an adoption cycle lag before it does.
In mature markets its common for people buy anything from really personal fashion accessories shoes & apparel to durables like washing machines based simply on product information coupled with a large number of genuine user reviews.
While the retail experience is still important, in relative terms the online one takes a primary role there.
So, till the buying behavior slowly shifts to a primary role of online, the marketing bucks will get spent offline where they can drive preference and sales.
But the mix of Online-offline is undergoing an increasing adoption of online to reach the end users and buyers in increasingly relevant ways, with sharper messages on digital platforms where all of us are spending a larger chunk of our time.

I sincere gestimate, after 3 years of this question been put up, would be that now in 2016 the dital spend for BRANDS in ME would be atleast 25to 30% of there total marketing budget.

With rest of the marketing mediums stagnant or degrowing, print media itself has taken 5-7% of down sizing.

Since the target audience have moved online, the marketing spent has to move towrds online. 

useful question. now a days social media marketing and digital marketing doing a great job and making the business for companies very easy. it has already been caught and it will go further because its an easy way to do business.