The technology that made the Internet possible was almost exclusively private sector technology. The desire, effort and vision to put together large networks was already in place by the time the government made their move.
The government helped, no question about it, but all the pieces were already there to make it happen. The private sector was driving that direction and would have built out very large networks regardless of the government.
As an aside, all of the above is typically covered in the first week or two of a networking course (I like Brighten Godfrey's [2]).
[1] http://ccr.sigcomm.org/archive/1995/jan95/ccr-9501-clark.pdf