Sure you can, since “common carrier” is a centuries old common law classification that the FCC has nothing to do with, as well as also a statutory classification under the Telecommunication Act inspired by the common law classification, as well as a statutory classification under other laws.
And even for the classification under the Telecommunication Act, one could argue the definition in the Act whether or not the FCC has applied it in a particular way (this requires overcoming the deference that courts give to executive agencies delegated decision-making powers by Congress, but it is not an impossible argument to make in principle.)
Explanation for why I downvoted your comment:
Your comment is completely unconnected to the statement above it. It's only connected to the story in that Cloudflare was involved with both, but if you're going to try to connect it to this story your comment should go at the top level. But connecting a highly controversial subject to this one by posting inflammatory comments does not seem to be a good way to connect these two different stories.
Cloudflare wants to project the idea of being critical Internet infrastructure without the governance and responsibilities that go along with that.
No one wakes up at a root DNS server one day and deregisters a domain because they feel slighted by the content of that domain.
If cloud flare hadn't effectively given up its status as a common carrier, by blocking stormfront, maybe it would have been able to defend itself from the sci-hub court order.
Cloudflare being forced to take down content in the future is exactly what people predicted when they decided to give up their status as a common carrier.
In related news, I did not give up my status as a fish by breathing air.
They are trying very hard to be a common carrier, because they want the legal protections that it provides them.
But, because of their actions, they've lost the court cases, and are unlikely to recieve that status.
EDIT: I'm unsure you can argue, with a straight face in front of a judge, that you're a common carrier when you've publicly blogged about determining to terminate a user because you felt like it. I'm not judging Cloudflare (anymore), just observing the pickle they've put themselves in because of their own decisions.