No, not at all. There was a lot more to the internet than just the web which was a later arrival to the net. The internet had been around for years. TCP/IP had been standardised for a decade or so.

http was just one of many protocols, and should never have ended up being used for everything, it's not suited. There was plenty of reason to be online years before Mosaic too. They just didn't yet have pretty pictures, ads and tracking built in to every request. :p