The case was specifically about a full page ad that the New York Times had taken out that contained a number of factual inaccuracies, such as the number of times King had been arrested during the protests, what song the protesters had sung, and whether or not students had been expelled for participating.
GP asked:
> facebook as a platform is not liable for information it contains. However if Facebook would publish fact checking info, that is wrong or missleading, are they liable?
> Could I sue facebook if their "facts" were proven wrong, and cause me damage
NYTimes v Sullivan held that unless the "facts" proven wrong are specifically about a private individual, you can't sue for damages if a newspaper publishes information that are factually incorrect. If the NYTimes Editorial Board published an article that claimed that the sky is green, you can't sue them for that. More practically, if the NYTimes Editorial Board published an article that "climate change isn't real", you can't sue them for that either.
Likewise, if Facebook published fact checking info that's proven wrong or misleading, they won't be held liable unless the fact checking info directly defames a private individual.
Also note that this is US-centric. Outside of the US, this kind of publisher speech protection doesn't exist, and such nations can cause legal problems for Facebook. Either FB would have to hide the contents from citizens of those nations, or stop operating there altogether.