Often, no. There's a relevant specific fee-recovery mechanism [0] in some, not all, states, because this class of lawsuit acknowledged as being problematic, as being an effective way to silence 1A-protected speech. The deeper problem is that litigation is extraordinarily expensive in the US, and generically easy to abuse as an intimidation weapon—for rich entities to use against those who can't afford lawyers; or for fanatics to use against normal risk-averse people who don't want lawsuits–don't value whatever it is the fanatic is fanatic about, highly enough to go to court to argue it.

[0] https://en.wikipedia.org/wiki/Strategic_lawsuit_against_publ...