The "internet" is a term used to describe connected devices which use a common networking protocol.
The "web" is the domain/namesever www. set of text pages available for access on the internet.
The "Google-verse" is what has become of the web and the internet in which the accessible sites are those that play the game Google created.
So, to answer your question, without getting stuck in the weeds, yes. You can create a new "internet" without Google.
Will anyone want to use it? Well... that depends on how you create it.
The ability to leverage a new Web depends on how well the innovation incentivizes adaption and thus creates an exponential network effect.
Search , !=, =\=, .NE. the internet.