This may indeed be a First Amendment requirement, but the court hasn’t had occasion to reach that question, because it just hasn’t arisen often enough. Historically, defamation of the dead could give rise to a criminal prosecution — the prohibition was generally justified either on moral grounds, or on the grounds that libels of the dead can lead to fights or even duels — and some state statutes still authorize that. But I don’t know of a single such prosecution in the past 30 years; I doubt that such laws would be viewed as constitutional today; and these laws have not eroded the common-law rule that there is no civil liability for defaming the dead.
If such a statement was made when Prince was alive, he could win a libel lawsuit if the Enquirer knew the statement was false, or knew it was quite likely false but printed it in reckless disregard of its likely falsehood. That’s the legal standard required for a living public figure — such as someone of Prince’s fame — to recover in a libel case. But now that he is dead, civil libel law no longer applies, however much the statement might upset Prince’s relatives (or even if the statement economically damages the estate and thus the relatives).
Note, though, that a statement about the dead might also defame the living, and the living can sue for injury to their own reputations. Thus, for instance, saying that Don, who is dead, was a criminal is not civilly actionable, even if it’s an outright lie, and even if it understandably upsets his son Stan. But if the statement is that Don is a criminal and Stan was his co-conspirator, that defames Stan, and Stan can sue for the injury to his own reputation.
See also this article in The Wrap, which reaches a similar conclusion.