Democracy Could Exist in Both Private and Public Organizations
Certainly, both states and non-state organizations could be described as being "democracies". This view is laid out by the Stanford Enyclopedia of Philosophy, which describes democracy as:
[Democracy] refers very generally to a method of group decision making characterized by a kind of equality among the participants at an essential stage of the collective decision making
In a general sense, this could be applied to any organization. That isn't very interesting, because we aren't here to talk about the general English-language usage of the word "democracy". In political science and political theory we are interested in politics.
Politics is About Public Concerns
Really, the first question about politics is "what is it?". The root word is polis - the ancient Greek word for "city" (See this page, published by Syracuse University). The city was the political unit in the ancient world. With some extension, the polis is the citizens who can make political decisions - the political community. In the latter sense of the word, we can meaningfully talk about the politics of a state of any size.
The affairs of the political community are politics. The affairs of private persons, businesses, and voluntary organizations are not a part of politics (this is the public-private dichotomy). In this sense, within the scope of political science (or theory) all democracies are states.
Example: A corporation governs itself by voting, including elections of officers. In the generic usage of the word this could be called a democracy because they are resolving mutual concerns using equal voting (in this case, one vote per share). Within the world of political science and political theory it is not a democracy, because the voters and corporation are not addressing the larger political community's concerns.
If you are interested in terminology, this preoccupation with states is referred to as statism or sometimes the statist assumption.
Democracy is about States
This notion of democracy being all about states is common across pretty much all of political theory. Pick your favorite theorist. If they talk about democracy (and aren't anarchist), they are talking about states.
A few common ones are below, just to get this started:
- Social contract theorists (Hobbes, Rousseau, Locke, et al.) describe democracy as one potential form of a social contract. The social contract fundamentally outlines the public world and government, not private organizations. Locke makes this most clear, because he takes great pains to distinguish the governance of families from states.
- In the socratic tradition (See Aristotle or Plato) democracy is one of the possible regimes. Aristotle came to this conclusion by studying constitution; Plato based on his experiences living under different governments in Athens. Both authors use democracy as a kind of state.
- Bridging the gap are republican authors, such as Machiavelli. Both in The Prince and Discourses on Livy he distinguishes democracy from other regimes (such as tyranny). He draws case studies from various states of his day and the ancient world, which also focuses his work on states.
Additionally, there is a political science journal called the Journal of Democracy. If you browse their articles, you will see that they are dealing with issues related to states - not the governance of non-state organizations.