I'm not sure whether my answer will be of relevance to your underlying niche-protection question. In my edition of choice, healing isn't a niche that's restricted to the Cleric, or Divine classes, or even Leaders. The Cleric, as the Divine Leader, is arguably the best at it, with both the widest array of healing abilities and – to the best of my knowledge – the most potent ones, but when there are Martial, Arcane, Primal, and Psionic healing abilities in the game, it's hard to argue that granting anyone in particular access to healing is encroaching on the Cleric's territory. Which isn't to say that the Cleric is without a niche, or that that niche shouldn't be protected, just that that niche is far more specific than 'healer' or 'divinely-powered healer' (the brief description of all of the classes on page 52 of the 4E PHB doesn't mention magic or healing at all, but describes the Cleric as A divinely inspired warrior). Note that despite plenty of other classes having access to both healing abilities (both magical and non-magical) and divine magic, no character who hasn't multi-classed into Cleric has access to any of the Cleric's specific healing abilities, many of which grant greater amounts of HP than same level non-Cleric healing abilities.
And, regardless of edition, it seems to me that it's more a matter of how the gods are being handled in that campaign than anything else. If it's a campaign where the gods are active interventionists, constantly granting miracles and blessings, being the direct and indisputable source of the powers of divine magic, and so on, then the answer is inevitably going to be different than in a campaign where the existence of the gods is something to be taken on faith, where divine magic is an observable fact, but the true nature of it is a mystery ... or in one where (whether the players are aware of it or not) the DM has decided that there are no gods at all, and all the world's religions are worshipping either things that are unworthy of their worship or which don't even exist.
In 4E, stabilizing the dying is a Heal check that can be performed untrained, by anyone. The absolute worst a by-the-book 6th level fighter can have as a heal check (assuming no penalties imposed by temporary conditions) is -1, which gives a success rate of 25%. And to be that bad requires you to have used Method 3 (rolling) to generate scores (something I can't recall ever seeing in a 4E campaign), and managing to get a 3 on 4d6 drop the lowest, and then deciding to put that in Wisdom, despite the game specifically labelling Wisdom a key ability for fighters and there being class abilities that make use of your Wis modifier. A point-buy fighter can't have a worse than 40% chance of success, and a standard array fighter's minimum success rate is 45%. At higher levels, the chance of failure almost entirely disappears, thanks to both the half-level skill modifier, and the bumps of +1 to all ability scores at levels 11 and 21.
Of course, the absolute worst-case scenario not involving some imposed penalty to the check would be to make the fighter level 1, losing the half-level bonus. Then our absurdly unwise (but very devout) fighter's heal check would be at -4, with only a 10% chance of succeeding.
In the unlikely event this scenario occurred in my game, I'd probably ask the player of the pious PC to roll a Heal check. If they want to treat a success as divine intervention, I'll neither confirm nor contradict their interpretation. If they opted to not roll, or failed on the roll (or rolls, because this could conceivably take multiple rounds to resolve), then it's on the player of the dying PC's death saves, and again, if they choose to interpret success as divine intervention, that's fine.
Now, there's a possibility that the player might reject the Heal check proposal. Perhaps they're a purist about things doing what they say on the tin. Their character isn't attempting to heal their wife with their skill at healing, they argue, so there's no reason for them to make such a check ... in which case I might ask them to roll a Religion check and on a success (level appropriate easy DC) inform them that while praying they've recalled that Pelor's directions to his followers include Alleviate suffering wherever you find it and Bring Pelor's light into places of darkness, showing kindness, mercy, and compassion and that sitting around praying instead of trying his best to alleviate his wife's suffering isn't the way of Pelor: Pelor is more likely to answer his prayers if he accompanies them with honest efforts to follow those instructions.
I might also try afterwards to subtly guide the player into realising that a truly devout follower of those teachings would probably at least consider getting some proper first-aid training ... or I might just leave it to them to figure out on their own. Though really, if they've been that active in bringing their character's piety into the game, there should've been plenty of opportunities by that level to show that followers of that religion seek training in healing as one of the ways to better serve their god (as it certainly seems to me that they would).
And setting questions of piety aside, if you've not got a reliable healer around (and some sort of back-up for when that character is the one that needs healing), opting not to train in even the most basic of healing (and also not carrying any potions or other healing resources) shows some pretty low Wisdom. An experienced fighter, regardless of religion, knows that people get wounded on the battlefield, often far from the best medical aid. Kudos to them for roleplaying their low stat, I guess.
(Tangentially, I still consider it odd that Clerics get magical healing abilities regardless of their deity of choice: to the best of my admittedly limited knowledge, most real-world pantheons have a god or goddess who is the patron of healers, with few if any of the others having anything to do with healing at all. The universality of magical healing among Clerics seems to me a relic of the fact that, despite D&D generally assuming pantheonic settings, the Cleric is fundamentally inspired by monotheistic concepts.)