Obviously, based on you even asking the question, it must be true that not all witchcraft in Supernatural requires demonic worship.
In fact, magic itself in the series is just one more mystical tool available for those that know how to use it. Much like the various weapons can be wielded by anyone, even if they are not its intended wielder (e.g. anyone could kill a demon with Ruby's knife, not just demons), anyone can wield magic if they know how to do it.
However, not all magic appears equal in the Supernatural world; if we take a look at the kinds of magical events that happen, we can group them into a few categories.
Witchcraft
This is most of the magic we see early on in the series; humans who have made deals with a demon in exchange for access to magical powers. Note that these people are able to produce varying magical effects simply by willing it to happen, but they also employ traditional spells, as well as totems and charms imbued with permanent effects.
It appears that this kind of magic does require supernatural intervention for a mortal to achieve, typically demonic intervention, though I supposed it would be possible for some other powerful being to give a mortal access to this kind of power. The items that witches create, however, appear to be usable by anyone (the boys steal hex bags from witches and carry them around, for example.)
Mechanical Magic
This is the type of magic we see non-witches using that involves spells and ingredients and incantations. In this type of magic, the effect comes from just "following the recipe", so to speak. Anyone capable of recreating the exact conditions needed by the spell can make it happen. However, they are limited by the precise constraints of the spell, and we've never seen any indication that a non-witch can "create" (or possibly "discover") a new spell on their own.
In this case, the magic is in the knowledge. Just by knowing the list of ingredients and symbols that Balthazar used to send Sam and Dean to the "real world", Dean fully expected to be able to recreate the effect on his own. This form of magic simply is -- its part of the world that is available to anyone who knows. (And is not part of the "real world" since it didn't work for Dean as expected.)
The incantations used in this type of magic sounds like it's supposed to be Latin (though modern TV almost never gets classical Latin pronunciation right -- it's always Chuch Latin at best). Even the exorcisms that Sam and Dean occasionally perform can be lumped under this type of magic, though arguably their "power" may be more divine than mechanical.
Enochian Magic
Much of the magical effects we see in the later seasons is Enochian magic -- the language of heaven. The sigils that the Winchesters use to hide from, banish, or trap angels and demons are largely Enochian. (If I had to guess I would assume a Devil's trap is probably this type of magic, though I don't think it's ever stated.)
Also, some of the incantations that angels like Castiel use are Enochian, which in the series has a distinctly different sound than Latin. (Side note, Enochian is a "real" language, in the sense that the Supernatural writers didn't invent it, but borrowed it from the guys who did.) Again, as the language of Heaven, being able to speak it allows one to produce divine effects that would be well beyond the range of mortals.
In my mind, there's a subtle but important distinction between the type of routine magic performed via spells, and the Enochian magic performed by writing or speaking Enochian, but I can't really articulate the difference. The angelic magic seems to derive its power directly from the words (written or spoke), as if saying or writing something in Enochian makes it happen. For "normal" magic, the ingredients by themselves have no power, it's knowing the "formula" to mix them together that produces an effect.
(I admit that I have no other canon support for this than my own intuition from watching characters that have used both, e.g. Balthazar using "normal" magic to send Sam/Dean away, or Sam/Dean activating Enochian sigils to banish angels.)
Supernatural Creatures
And, of course, the non-mortal creatures in the series all have their own abilitys and powers that can occasionally seem like magic. These creatures also seem capable of using more mundane magical forms when it suits their needs. Castiel, for example, uses a normal spell to locate Anna, presumably because his own innate abilities were being blocked or otherwise unusable.