In the UK at least, these consequences (website going offline / certificate warning / unsearchable in the search engine) would likely be deemed "pure economic loss" following
Spartan Steel & Alloys Ltd v Martin & Co (Contractors) Ltd [1973] QB 27 and
Murphy v Brentwood District Council [1991] 1 AC 398 where the Court of Appeal and House of Lords respectively held that unless some sort of physical harm was suffered to you or your property, the losses were held to be "purely economic" and so not recoverable in tort.
It's unlikely that any claimant would be able to show a contractual provision that enables them to claim for damages against Google (thus allowing them to sue in contract), so a cause of action for tort would be the usual way to sue Google - except unless Google makes you suffer some form of physical harm or damages your property, you're unlikely to be able to recover any damages for your website suffering these consequences, in the UK at least. I understand US law may be quite different.
There's a testable argument to be made about the requirement for "damage" to your property (the website) being inflicted by the certificate warning, but policy arguments on the matter of "ripple effect" liability makes it seem likely the courts would hold that Google isn't liable.
Also Google is probably far better placed to weather lawsuits than most ordinary people; they can probably afford to induce the other party to settle out of court, and presumably the relevant monopoly and abuse of market position laws only allow a regulator to take legal action (the ordinary consumer being restricted to contract and tort lawsuits).