It does exist for the military and there are closed networks for businesses. But not a public internet for the masses.I find hard to believe there wouldn't be something like that even for the military or NASA.
From the For All Mankind wiki:
The Government Computer Network is a massive network of government computers all linked together, including the DoD, CIA, State Department, and NASA. It is highly regulated by the US government and only very few services are available for public use.
Basically, in this timeline, the US government only allows it to be used for an email equivalent and video mail. This might be a result of the Soviet Union's continued existence into the 21st century and the US government being shy about sharing the particulars of a technology that started out as a military program.