In Morrowind, it is rather obvious that the Tribunal Temple and Imperial Cult are, at least, partially based on magical instruction, given that both have several skills among their favored. This is also rather blatant in Oblivion, where the Great Chapels have at least one spellmerchant each.
When we come to Skyrim, however, it seems less obvious. There are no spellmerchants in the temples, nor are there any favored skills we can look to for an indication. On top of this, there has been a change between Oblivion and Skyrim, with magic falling out of favor after the Oblivion Crisis. The extent of this is hard to pin down, but with the fall of the Mages Guild, the only known institution with the goal of making magic accessible to the masses, I doubt magic is anywhere near as common as it was in the 3rd Era.
However, I think the temples in Skyrim, and likely beyond, are still practicing magic. Two of the Master Trainers in magic in Skyrim, Danica Pure-Springs (Restoration) and Hamal (Enchanting), are the leading priestesses of their temples (Kynareth and Dibella, respectively). Keeper Carcette, the Keeper of the Vigil (leader of the Vigilants of Stendarr in Skyrim) is an Expert Trainer in Restoration. Runil, the Priest of Arkay in Falkreath, is a Common Trainer in Conjuration. And Aphia Velothi (in Dragonborn) used to be a Priest of the Tribunal/Reclaimations and is a Common Trainer in Restoration.
It seems to me that magic remains an integral part of what it means to be a priest in Tamriel in the 4th Era, even in Skyrim.
Does anyone have more info on this than I could find?