Come to think of it, that might not be the worst thing in the world.
Something at a publisher level, where the publisher themselves is given a fake news rating, rather than individual stories, might work, but even that is full of difficult to solve issues.
Publishers always had reputation (good or bad) based on their work, which maybe could be displayed with a rating - the problem is when even those with good rep are wrong...
This is a practical suggestion, but not one I find reassuring. The Youtube copyright system works fairly well from the perspective of big rights holders, but is relentlessly abused for both censorship and corporate overreach in taking down fair-use or even non-infringing content.
That's basically the outcome I expect from a fake-news-takedown system, too. It would almost inevitably end up working for whoever was willing to throw the most time and money at manipulating content.
But I think there could be some mechanisms in place, similar to how Youtube handles copyright claims - where upon reporting, or a request for a source from any Facebook user - the publication would get it's Reach capped, or the publication would be hidden, until further evidence is displayed.
Now, there are a lot of problems with this approach - first it's user based (which has it's pros and cons), second it may be target for malicious activity and anti-competitive pratices - but on the other hand if the source was displayed, it would somehow protect them (at least briefly), without having the Reach capped, but still flagged if the source is dubious.