Not even remotely. Right now, the data that goes through the public Clouds™ is roughly 49%. Amazon itself is has the biggest piece of the cloud market share at 40%. And this is only from sources that can't control all of the data.
That in itself puts in perspective how much of the entire internet they don't store and that's not even going into how very complex the internet is: CDNs, DNSs, VPNs, P2Ps, the dark web, etc. might or might not be accounted for in several studies.
So no, they don't physically own most of the internet.