To be fair, and from an academic viewpoint, no institution is above fact checking.
Of course, that doesn't mean Facebook *is qualified to do so*, though I suppose they could hire qualified people.
Peer review only means so much and, well, I think people would be surprised with what goes into 'peer review'.
A quick look indicates that the BMJ has a solid impact factor (it's outside of my field), so that's a good indicator. But, fact checking is a fact of life and nobody is above it - nor is anyone beyond reproach.
*I* have done peer review. Yeah... Let that little nugget sink in!