The Internet was (and is) an evolving medium. It predated the web, with the first commercial Internet Service Providers (ISPs) beginning around 1989. Early features included e-mail, FTP (File Transfer Protocol for making files available to others), gopher (a hierarchal index of FTP sites and their contents), and newsgroups (open predecessor to commercial sites such as Facebook and Twitter).
There was a cross-pollination period between BBSs (Bulletin Board Systems, which started around 1978), private companies that offered their own computer offerings such as CompuServe and AOL, and the Internet.
The web itself could not exist without a browser. (Well, people have stretched other programs to include browser functionality, such as EMACS.) So the seminal date would be in 1990 when Tim Berners-Lee invented the web browser. He called his program WorldWideWeb. The first really popular browser was Mosaic, which came out in 1993 and developed by Marc Andreessen.
It is mostly incorrect to think of the Internet as an "American" thing. While the American government did start it off with Arpanet, that was restricted to a government/military/higher-education audience. Once ISPs started offering Internet access to anyone with a credit card, they didn't care what country you phoned in from. Plus ISPs opened up in a huge number of countries very quickly.