Absolute nonsense. This is not the case at all and has never been the case. The state has never been bound to enforce civil proceedings through criminal law. It would actually defy the very base claim of the state to allow civil courts to bind it to enforce their verdicts.
What exactly is a false accusation? It's speech, by the most basic definition of speech. Criminalized speech. It is the gov't saying "You cannot say this to this person." Does a swatter, using his speech in a way that directly leads to someone's death, commit a crime?
Again, absolute nonsense. Speech and non-speech are blatantly two separate things. Have always been adjudged to be two different things and even in cases in which speech is criminalized are fully and completely judged to be two separate things.
If not the state would need not pursue a charge for the act of speech, it could simply charge for the actual action and related damages incurred. Again, the state's own actions belay the underlying base of this claim by liberal theory.
Let's ignore your (actual, not RE variety) whataboutism by saying the state prosecutes the cop for the shooting as well. Do they prosecute the swatter? If so, what do they charge him with? If murder, what action did he take? What is the criminalized action that he did?
A consequentialist (who may or may not also be a determinist) would answer that foreseeable consequences are not unintended. In terms of using the state as a proxy to commit violence against their fellow citizens, I may fully agree with them.