I'd get it if registering your domain with GoDaddy was a big visible thing that you might want to keep your name away from controversial content, but honestly I have no fucking clue who's hosting the domain of what or who's registered where.
I mean I don't exactly hunt down the DNS information of everything.. heck anything I look at.
Honestly who cares.
I don't think that can actually happen. The first amendment is a prohibition against Congress passing certain types of laws, not a cause of action for citizens to sue people. You have no right to freedom of speech or expression on Hacker News, for example. You are also free to contract away that right with GoDaddy or anyone else; see NDAs.
The only liability GoDaddy could have are breach of contract, if the contract does not allow this (which it likely does), and a dispute with ICANN over registration rights.