DISCLAIMER: The following post contains massive oversimplifications.
Er… I thought the web was the internet!
The Internet predates the Web by a couple of decades (late 60s vs late 80s).
The Web is specifically the text and pretty pictures what you see in a web browser. The Internet is the underlying infrastructure. For example, without the Web email would still work. Without the Internet nothing would work.
And no, my anxiety levels are off the scale anyway so try to reassure me, not worry me.
The proto-Internet was a coming together of Academia and the Department of Defence, it’s underpinnings were quite literally designed to withstand a nuclear strike. The short answer is, it’s not going anywhere.
The Web is a different matter because it’s not really a single entity (not that the Internet is either) but rather a series of distributed servers and supporting technologies, distributed across the Internet. To take down ‘the Web’ you’d likely have to find some sort of vulnerability common to Web servers in the West. As different servers use different technologies it’s unlikely that there would be some sort of one-size-fits-all magic bullet; a vulnerability in Apache 2.4.1 might not exist in Apache 2.4.2 and almost certainly wouldn’t exist in Microsoft IIS servers.
It’s entirely possible that something might knock out web servers on a large scale, or similarly cause major Internet traumas. It’s possible – likely even – that specific organisations might be targeted. But taking out “the Internet” or “the Web” is about as likely as eradicating French overnight.