Well your question is somewhat vague. What exactly you mean when you say "completely social"? Do you mean that the content of the web will be completely social constructed? Or the technology supporting online activities will catalyse the web? Or are you referred to the interconnected / networked / interoperable / collaborative nature of social media?
The web (i.e. the cyber-structures that is comprised by) has become an extension of the actual (offline) world. Does the offline world is completely social? Is every artifact (i.e. handmade object) socially constructed? Activity theory argues about the socially-situated phenomena of human activity where objects (or objectives), mediating artifacts such as tools and signs, actors, and sets of rules, practices, etc. are part of the activity system. but how the rules or practices are being established? How the tools are created? Does this answer that every human created object is socially constructed?
I believe that what is called the social web was an inevitable advancement. Humans are considered social beings and always are seeking ways to communicate, to come close and create communities for their own benefit. So probably the web and any other human creation will be as "social" as it can be.
The question should be "what can be considered as social, what is web and what is social web?". First of all I would distinguish between the word itself (meanings) and then as a phrase in connection to web. It will help you to find out how social web is. On one hand you can see web 100% social and only social, but on the other hand, as a virtual world, it can never be concerned as really social. Question about socialization of web is pretty deep.
Well, 'social is as social does' and it rather depends on what we, as 'prosumers' make it through networks rather than on ownership and control for commercial purposes. See Yochai Benklers 'The Wealth of Networks ' on this if you haven't already. Just Google it and download free from his own site.
I think web is becoming (if not already) a platform of human activity.
when we want to find the forex for a currency, we don't wait until the 7 o'clock news, but we do a few taps on our mobile device, which pull data from the web, and the information presented before our eyes.
when we curious how our nephew is doing with his school, we sit on the couch, grab the tablet and check out their Facebook page or the family's circle in Google+, streamed down from the web.
when we walk down the street and remember to buy flowers for our loved ones, we taps, and that box on our hand send information of location, workout the closest flower shop and show us the way there. and afterwards we can give feedback when we are happy (or not) with the service they provide, all streamed down from and up to the web.
and much more..
web is (or at least increasingly becomes) the platform for human activities; if the activity system it affords is social, then web is social; if the activity system is non social then web is personal.
human is a social being, God created so in Genesis; and Aristotle re-articulated it when he said we are by nature part of a community (note: not exact citation).
So with the continuous proliferation of mobile devices and ubiquitous computing, we can access the web from anywhere, anytime, through any ways; if humans uses the web according to human's nature (most likely is), then humans will use it to socialized (both for better and for worst). thus to answer your question, IMHO yes.
Of course the quality, intensity, degree, (or whatever means of measurement) of the social interaction would varies as Tereza pointed out.
For years, experiences were lived largely “offline.” Our interactions have been in person, in front of a television, or through headphones. But times are different. Internet technologies and social media have enhanced our online experiences. We enjoy interactivity, video, audio, and pictures just as much from computer screens as from offline experiences.
We want to learn, share, and interact from the comfort of our mobile devices more than ever.