For absolute certainty, you could do what the court in the linked Chat-GPT case did: contact the clerk of the court that issued the alleged opinion and ask if that opinion really exists.
In the case of cases reported in the various leading case reporters, however, free Google searchable services will generally have full text copies that can be found online. These resources have close to 99% of published appellate court opinions that are reported in recognized case reporters in the U.S. Secondary materials are often hard to get online for free, but primary legal authorities are usually easy to find.
Westlaw, Lexis, and some specialized subject matter services like tax publishers, often also have unpublished or unreported decisions that would not be revealed by such a search. But, while the vast majority of cases decided are unpublished, the vast majority of cases cited in legal briefs are published decisions that are reported in recognized case reporters. And, in some courts, citations to unpublished opinions, must be accompanied by the full text of those decisions or are entirely forbidden.
So, you could restrict inquiries to court clerks to cases whose existence is not verified from free sources on the Internet.