Internet was neither the only computer network available to consumers, nor the first one. Users had been pretty connected even before they started having IP addresses. Also, Internet is not a synonym of World Wide Web.
Most consumers didn't have computers. Let alone modems or any online service. Consumers were not pretty connected when Doom was released and wouldn't be for until the end of the decade. Businesses and schools with Internet access were 1) decidedly not consumer systems and 2) not the target market for video games.