I definitely appreciate the time you take in explaining your point of view. Thank you.
However, you are coming from a 4e POV, I'm coming from an 1e POV, on a 5e board.
With 1), nothing. Hit points are, have been, and always will be nebulous as to what they represent. Your comment that use of the Religion skill to sooth and calm in no way implies it is a direct path to healing, and in fact is explicitly a lore skill. It does unlock some abilities if your are trained in it, which are enumerated and specific.
With 3), vague alignment. They're not divine, but mundane, however. Let's actually look at your statements.
Gods don't answer mundane prayer/recitation/ceremony.
Of course they do. The numinous powers do as they will.
Mundane prayer/recitation/ceremony doesn't restore hit points (and its proxy or component part in 4e; flagging spirits, resolve, steadfastness).
It might, but then it's not divine. It's mundane, which isn't divine, or arcane, or primal, &c.
This is doublethink. Is it mundane, or is it divine? You want it both ways. If the divine is answering a prayer, it's divine action.
Does the fighter find a space of calm within and gather strength to him, boosting his resolve? Sure, no problem. It's not divine and there is no prayer answered. It's all internal, regardless of result.
Does the fighter make an earnest and passionate plea to the numinous, and find he or another benefits from some benevolence? Sure, no problem. The divine heard the call and responded. But, it's a divine action and not mundane.
This line of discussion is wholly orthogonal to the OP's question. Because, even with the mundane, yet divine, but really still mundane non-magical unconnected to the numinous Faith Healing daily you mentioned, the answer is still "the fighter doesn't need to make a Religion check". Because there aren't any rules for Divine Intervention in 4e as customarily defined in other editions. He just uses the power that the player had the foresight or theme to choose.
If it is supernatural action, then it's supernatural action. I can accept that warlords can inspire and motivate other to "ignore wounds" to complete the task at had. But that's not supernatural, it's mundane, by definition. (Well, "martial", which I don't think was ever defined much more than "action movie" logic.)
Here's the question, again:
How do you answer this in 5e? In 4e? In AD&D? There are no Daily powers left, the Encounter powers are consumed. No time for rests, Long or Short. Potions are broken, seeping into the cracks on the floor.
There is no defined answer except at the individual table level. Despite our disagreement on whether or not some powers are divine, mundane, or whatever in 4e, this situation is undefined. Why are we arguing about 4e? If you want it to be the player makes an earnest prayer and the character tries for a DC: 20 Religion check, with a success allowing the target to auto-stabilize / spend a healing surge / gain 1d3 hp, that's great! But, that's not in the book, it's a decision for and by the table.