Take the following character: a human fighter of moderate but not spectacular level (let's say he's level 6 or so). He's good at fighting but also a very pious man. He worships the local NG deity (let's say Pelor or some close equivalent; the most common NG deity who includes healing among other things in his sphere of influence), and frequently roleplays his prayers to said deity. He buys potions only from this deity's temples, and leaves a little extra gold as a tithe. He frequently helps other worshippers in need. He spreads the faith. He never asks for anything in return.
However, he only ever takes levels of fighter, and has no spells or supernatural abilities of any sort. A few ranks in Knowledge (Religion) or whatever the system calls it, perhaps, but he's not a cleric, not by a long shot. His Wisdom is low enough that he couldn't cast spells even if he wanted to learn. He has no interest in being anything other than a fighter.
Now, he's fighting a pitched battle, and his character's wife, a fellow PC and likewise a faithful follower of the same deity, is felled by an orc, being dropped unconscious and helpless and down to within a point or two of dying. She might stabilize, but she'll probably die soon without help. With no healer in sight, the fighter finishes off the orcs and rushes to his fallen wife. Not knowing whether she's alive or dead and possessing no healing skill or magic items to speak of, the fighter prays for his wife's survival, begging his NG deity to heal her and keep her alive.
What happens next?
Pick whatever option you like the best.
This is a very interesting scenario. Is it based on something from one of your games, or is it strictly hypothetical?
So, the thing about D&D specifically that makes this scenario weird is that 'faith' has a mechanical component (sometimes! not always). There are specific classes that use faith to cast what is basically magic. Of course virtually every PC or NPC in the game follows a god of some sort without being shoehorned into a divine class. So arguably piety does not REQUIRE the PC to be a divine class.
I see several responses along the lines of, "Well, if the PC is so pious why isn't he a cleric?" But there are lots of examples of pious non-divine characters in D&D (the classic dwarf fighter who is a devotee of Moradin being an obvious example). The other problem is that a cleric isn't just a fighter + piety. A cleric fills a different niche from a fighter entirely.
A paladin is closer to being a 'pious fighter' but even here there are key differences. Depending on the edition, a paladin may have important mechanical differences from a fighter (probably including heavier armor). Paladins are also notorious for having alignment restrictions and following certain very specific play-styles. This may be off-putting to some players.
So it seems like a player might have legitimate reasons for playing a pious fighter.
So the next question is, does our pious fighter get to enjoy even a small taste of the abilities reserved for 'mechanically pious' classes?
I say 'yes'. The only reason to deny the fighter his miracle is because it supposedly infringes on the territory of the cleric. But we're not talking about letting the fighter go about casting cure spells willy-nilly. This is a one-time, limited use event. There is apparently no cleric in the party to feel affronted by it. So basically it hurts no one and it is an awesome payoff for the player's character. And this player has obviously been diligently roleplaying his PC, so I think he's earned that payoff.
I would have the payoff take the form of a cure spell out of the blue, unless the DM had story reasons to have one of the gods put on an appearance. Either way, it should be something visible, miraculous and more interesting than a fudged dice roll.
Now, going forward I would talk to this player about steering their character towards a class with mechanical divine casting abilities. Probably paladin, because everything about him screams 'pally' to me! Perhaps the PC has a divine vision and is given a quest, upon completion of which they are transformed into a paladin. That seems to be the direction their character arc is taking, and it would let the player have a real mechanical benefit for his character's piety, which ultimately I think would be good for everyone.
In other words, say "Yes, and..."!
"Yes, your prayer was answered, and you also see a vision calling you to become a paladin!" Problem solved, story advanced, rules of the game appeased.