The site can ask for such a setup, but Google is very much in a position to refuse and just don't index that site, unless it's one of, say, top 1000 sites.
I don't see any legal ground for such a limitation, so I can't imagine what would e.g. eBay or Walmart put in their suit if they tried to put such a limitation and then saw Google not honor it. Maybe someone with a legal background could comment.
There isn't currently a legal ground for that (in the US). The context of this thread, though, was to propose that such grounds ought to be introduced by the legislature.
I can see the rationale but at some point Google needs to make money on what it contributes to the picture.
On one hand, Google is in large part responsible for the idea that they’re just a free service and anyone can use them without payment, so they’ve trained users to in essence think of them as a non-profit utility which they’re obviously not.
On the other hand, how do you allocate Google’s value when no-one has ever paid for it or been asked to pay for it?
You can legislate Google into the ground (unlikely, but theoretically possible), but what would replace it? Any other similar service operating at their scale would need to make money too.
I don't see any legal ground for such a limitation, so I can't imagine what would e.g. eBay or Walmart put in their suit if they tried to put such a limitation and then saw Google not honor it. Maybe someone with a legal background could comment.