Per the Thrall-focused novels from the Cata-MoP era, its largely from moisture/water in your surroundings. Although people often go for the “just summon from the Elemental Planes bro” angle, we virtually never see that being the case for mundane shaman calls.
Every time people directly tap into the Elemental Planes, bad things happen. Especially where fire elementals are concerned, as its been reiterated a few times now that they hate mortals.
This is further reinforced in the Darkspear Rebellion arc when Garrosh had the Dark Shaman prematurely corrupt the elements around Durotar and the Barrens to counter normal shaman from calling upon the elements in their surroundings. If it’s as trivial as just tapping into the Elemental Planes, this plan wouldn’t have worked.
But it did.
No. As we see in the Vanilla shaman class quests, it’s not water itself that performs the healing, but the Spirit of Life. We see later in the Thrall-focused novels that shaman heal by calling directly on the Spirit of Life quite frequently (in fact it’s the primary form of healing!) What the Vanilla shaman class quests teach us is that water is naturally rich with life energy, thus making it a necessary function of life.
Water is fuel through which shaman release the life energy, and it’s (usually) a resource that’s readily available. Mages would have to learn to manipulate the spirit of life itself to heal; water by itself doesn’t do much more than get the person wet. Although not an impossible thing (see Tidesages; a mage/shaman hybrid), you wouldn’t be purely a mage at that point anymore.