Yeah. There is also a right way and a wrong way to do it. Twitter is never the "right" way.FIFY.
In most reasonable orgs, you can get away with contradicting your supervisor. In a reasonable org that results in a dressing down to a write-up/warning.
However the penalties tend to increment exponentially as you move up the food chain. Correcting your boss is FAAAAAAR less devastating that correcting the CEO.
He deserved it.