If it would then I would fear it like the plague. Being able to seek legal redress is so fundamental to a free society no matter how repugnant the litigants are. Besides making it illegal would merely push these activities under the table. At least this way we know who the players are.
Useless/worthless lawsuits are a plague. This has less than nothing at all to do with "free speech"!
The First Amendment deals with the press...NOT with citizens and most assuredly NOT about spurious lawsuits.