There is a category of muktzeh on shabbat called Nolad. Nolad means literally, something that is born on shabbat. It is interpreted in the modern era, as meaning anything which you did not have access to before shabbat, you can not make use of on shabbat.
Mail for example (in countries that deliver on shabbat) is considered Nolad. An egg laid on shabbat is a classic example.
If we found a way to access the internet without breaking shabbat, (through some advancements in technology) would the content accessible through said technological advancement be considered nolad?
My rationale is as follows.
Facebook is updated by the minute, so social networks have new content from people.. what you are reading was likely written within the past 12 -24 hours.
Websites are generated dynamically. When you load a page, it instantly calculate which ads to show, which content to display etc. This layout and what things you see from second to second, as a single unit, I would argue is nolad.
When you visit a website, it updates your cookies, or the page hit counter, which update and are displayed with that new information (tracking, or on the site itself)
The programs existed before shabat, as did most of the content, it is only being displayed differently.
You don't physically touch the internet, so it can't really have a muktzeh status.