Earlier quoted context omitted.
The order does not say that they don't have the right to fact check. It simply says that they don't have the right to fact check and enjoy section 230 immunity.
FoxNews.com has comments on their website. By the logic of the order, Fox News should also lose their immunity and be liable for suit for anything posted by commenters to their online articles.
There was a later ruling in Hungary that seemed to go the other way though: https://globalfreedomofexpression.columbia.edu/cases/magyar-...