I don't think OP is asking if Bitcoin can physically and technically be bigger than the internet, he's asking if Bitcoin will have a larger, more profound effect on society like the internet did.
I think by 'bigger' than the internet he means it will change society more profoundly than the internet. It will change how we bank, how we work - it will continue to mint millionaires, shift power, open up possibilities that were not there before. I agree.
It's just semantics, but isn't any impact that Bitcoin has by extension attributable to the internet as well? After all the internet is what is enabling Bitcoin in the first place. Did anyone ever say that email will become bigger than the internet? Or that the world wide web will become bigger than the internet?
Maybe he meant to say that Bitcoin will become bigger than the world wide web. That would still be a contentious statement, but at least it would make sense. Saying that Bitcoin will become bigger than the internet, however, is a meaningless statement that only makes for a nice headline and nothing more.