Short version: Yes
Long Version:
The internet started out as a ARPANET (Advanced Research Projects Agency Network), a project by the United States Deparment of Defense to facilitate quick communication between remote computers. It was based on the idea for a 'Intergalactic Computer Network' by American computer scientist, J.C.R. Licklider, in 1962, who would also go on to lead the development of ARPANET. During the development of ARPANET, the two basic concepts that make up how the internet works were created and developed: Packet Switching and TCP/IP.
Packet Switching is the method used for breaking down data into 'packets' to more efficiently transfer them over a network. It was principally invented by Polish American scientist, Paul Baran, based upon the mathematical concept of 'queuing theory' developed by American engineer Leonard Kleinrock a couple of years earlier. The term 'packet switching' was coined by a Welsh scientist named Donald Davies, who coincidentally developed a similar method, a few years after Baran first published his own. While Baran invented Packet Switching first, the name that Davies gave it, was more catchy, which is why we call it that today.
TCP/IP stands for Transfer Control Protocol and Internet Protocol, and it is basically the set of rules for how computers talk to eachother. TCP/IP was invented and developed by two American scientists named Robert Kahn and Vinton Cerf for the ARPANET project.
While there are many people involved in the development of the internet, these are the key figures, who are responsible for inventing and developing the key concepts for how the Internet works
ARPANET switched online on August 30th 1969, making that the birthday of the Internet.
Originator: J.C.R. Licklider
Packet Switching: Leonard Kleinrock and Paul Baran as the originators, and Donald Davies for coining the name 'packet switching'.
TCP/IP: Robert Kahn and Vinton Cerf
So in short, yes the Internet could rightly be called an American invention.
The Internet is often confused with the World Wide Web, created by British scientist, Timothy Berners-Lee and Belgian scientist Robert Cailliau in 1989. The WWW is a system for accessing documents via the Internet, using the concepts of HTTP and HTML. It is basically a more user friendly way of accessing and displaying information on the Internet, but it is only an application that exists on top of the Internet, and it is not the Internet itself, which had already been around for some 20 years prior. Some other commonly used Internet applications are E-mail, File Transfer Protocol and Internet Relay Chat. A once popular application was 'usenet' an early type of message board, that lacked the central server of modern day web forums, to whom usenet has ceded its popularity over the years.
Sources: http://www.livinginternet.com/i/ii_arpanet.htm
The internet.
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