The rules as written just don’t cover the case of a non-spell (or spell-like-ability) attempt at magical healing. Most places where a caster level check is necessary (dispelling, spell resistance, etc.), a supernatural ability would not apply (they cannot be dispelled, they ignore spell resistance), but this special case does not appear to cover them at all. Even the wording itself is sloppy:
any attempt to heal [...] the spell
The sentence itself implicitly assumes that the attempt to heal was via spell, even though that may not be the case.
Notably, the 3.5 version of the horned devil uses different wording:
Infernal Wound (Su)
The damage a horned devil deals with its tail attack causes a persistent wound. An injured creature loses 2 additional hit points each round. The wound does not heal naturally and resists healing spells. The continuing hit point loss can be stopped by a DC 24 Heal check, a cure spell, or a heal spell. However, a character attempting to cast a cure spell or a heal spell on a creature damaged by a horned devil’s tail must succeed on a DC 24 caster level check, or the spell has no effect on the injured character. A successful Heal check automatically stops the continuing hit point loss as well as restoring hit points. The check DC is Constitution-based.
This makes it quite clear that the infernal wounds resist healing spells, not necessarily “any healing attempt.” On the other hand, it gets really specific about cure and heal and therefore fails to cover all manner of other healing spells, leaving them basically worthless—which would really suck for an undead creature with an infernal wound, as it must suffer from a cure spell in order to stop the continuous damage (and for some reason, RAW, that works despite the damage the positive energy is supposedly doing to the undead creature’s body).
The Pathfinder wording seems to have been an attempt to be more general, rather than specific to cure and heal, but mistakes were made in doing so. And it’s impossible to say whether it was intended that supernatural healing be resisted, or how that resistance works if that was the intent.
So basically, the rules, as written, are deficient, and the authors’ intent is far from clear. A GM ruling is basically all you have to go on. Personally, I would rule that wholeness of body (or lay on hands or whatever else) does just work, similarly to how Su abilities cannot be dispelled and ignore spell resistance.