It's a question of the content you put in the word "web". As a corrolary: I believe that "TV" would not exist without advertising.
I know that electomagnetic radiation, signal processing, and ray tubes are wholly orthogonal to Kelloggs selling cereal. But the modern television media landscape and its dedication to Brave New World style audience-capture is unfathomable without people splashing money at it to sell things.
So what is "the web"? Telnet, TCP, and HTTP? We don't need your stinking banners! We have oodles of excess server bandwidth and our websites are rarely more than 10K (of gorgeous hand-crafted fully-accessible HTML, natch).
But is "the web" your advertiser-supported newspaper? Your advertiser-supported searches? Your advertiser-supported social media news feed? Your advertiser-supported free phone app? Your advertiser-supported image host? Those things are, so far, only sustainably available through advertising.
Crack the micro-currency and micro-valuation code for everyone online and maybe that "web" can live without ads... we're not there yet though :)