There are people, a LOT of people, who literally think Google IS the internet. If they can't get to Google in a web browser, they think the internet is broken; "I can't get to my internet!" And they know of no other portal thereto.
Just ask some folks over age 55 or so. You'll find this user mindset is very common.
It's like Bear Grylls or some other "survivalist" who knows how to pick berries in the woods and make a shelter laughing at the idea that a breakdown in modern argiculture would have an effect on the world.
But people in general? I strongly doubt it. Hang out with some "non-technical people" (they will often helpfully describe themselves this way for you) while they use the computer sometime. If they didn't have Google, they might be able to get to Facebook, but that would be pretty much the boundary of the Internet for them.
I'm not sure why people here are acting as if Bing, Yahoo etc. don't exist and don't offer a decent service that would be adequate for the internet to continue functioning without Google.
Real World Example: Some relative of my girlfriend set up her mom's computer to use Yahoo. But she has since been convinced that Google is better and she should use that. So what does she do? Every time she starts up her web browser, she types "google" into the Yahoo search box, clicks the link to go to the Google homepage, then does her search there.
Yeah, if any major service that millions of people used every day disappeared, there would be a period of adjustment. But that's all... a period of adjustment.