The swarms of dead men fighting for the good side always seemed wrong to me. I know they did it to be released from their plight, but.... It's one thing to curse somebody, it's quite another to back it up. How did Isildur back up his curse on the oath breaking men of Dunharrow?
From other answers on SFF, we know that only Ilúvatar has the power to cause men to remain on Arda after death (from Silmarilion).
As such, the best guess is that somehow, the oath (or the curse) was "by name of Ilúvatar".
There are two elements in this:
- The Dead Men's original oath, and,
- Isildur's curse.
The assumption that Isildur's curse was the primary factor in the Dead Men's eventual fate may not be entirely valid. Oaths in Tolkien are powerful stuff. Perhaps the best example to compare with is the oath of the Feanorians:
They swore an oath which none shall break, and none should take, by the name even of Ilúvatar, calling the Everlasting Dark upon them if they kept it not ..... For so sworn, good or evil, an oath may not be broken, and it shall pursue oathkeeper and oathbreaker to the world's end.
The implication here is that the breaking of the oath was what actually sealed their fate, but Isildur's curse just made things a little more specific.
It is important to understand how important oaths and curses are within the context of Old English / Old Norse cultural traditions and JRRT incorporated that into his works. user8719 mentions the Feanorian oath, and that's one example. Curses show up in the case of Hurin and his descendants in The Silmarillion.
Also, check out the discussion of Elrond and Gimli on swearing oaths in support of the Fellowship. Elrond councils against it - recognizing the power of oaths and how dire they can be.
As a scion of a great and fated line of men, Isildur wasn't necessarily a wizard of any kind, but an oath sworn by him or an oath sworn to him and his kin would be especially meaningful.