Might not something like the Streisand effect take place? Right or wrong, vindicated or not, the effort would simply focus more attention on the whole mess.

As for the legal questions:

Remember that Kavanaugh is a public figure, so it is much harder to defame him under New York Times v Sullivan. It could be hard to win.

His damages would be hard to prove, which is always true in a defamation case. He got the job, after all. Which is why, in most states, certain kinds of defamation (called defamation per se, and this often includes defamation concerning sexual matters) do not require proof of actual damages. Damages can be presumed, but such damages will generally not be large.

The whole idea of vindicating the public interest is nice and all, but this is not really a good way to do it. Mainly for the first reason I mention - it will merely re-ignite the controversy.

Lots of problems can't be solved in a courtroom.