Amazon could pull a CloudFlare and knock them off the internet as nationalenquirer.com seems to be hosted on AWS. My guess is Amazon/Bezos already has multiple backups of the Enquirer's digital assets for forensic purposes.
even if that isn't the case, the amount of effort required for amazon to obtain a warrant from a magistrate judge or however they'd play it is negligible.
All they need to do is to tick one of the boxes in their ToS and kick them off their property. Heck, there's an entire sub-category of law colloquially known as "Amazon-law", where you hire folks who specialize in navigating the intricacies of Amazon's policies so your business doesn't get crushed:
https://www.theverge.com/2018/12/19/18140799/amazon-marketpl...
Edit: https://aws.amazon.com/aup/ TL;DR - they can do whatever they want.
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Character assassination of the chief head of AWS is damaging to the reputation of AWS in no uncertain terms.
We're at a point in time where the public, politicians and the tech community itself are becoming wary of tech monopolies. Currently Amazon has avoided the brunt of it (mostly due to being consumer friendly unlike Facebook) and the scrutiny it has had, has been focused on it's retail business.
Bezos booting someone off AWS would bring a ton of public scrutiny to an even more powerful monopoly, one that the public and politicians are by and large ignorant of. Plus it would (rightly) shake the tech community out of its complacency around cloud lock in and encourage us to explore solutions. (I realise there's already a lot of good work happening around this, but it's not an issue I've seen the wider industry talking about much)
Even if AWS has nothing to do with this it (unfairly) reflects extremely bad on them.