I think it is possible for majority rule to be, in an important sense, undemocratic. Imagine a society split 60/40 into two comprehensive factions, such that people in the same faction always vote together, and against the other faction. In such a situation, I think it would be misleading to describe a system of majority rule as 'democratic'. It is not the people (generally) who rule here; what we have instead is a mere oligarchy, however large: 'rule by the majority faction'.
What more is required for democracy, then? Total consensus is an unrealistic ideal, and democracy still ought to be possible in the face of robust disagreement. I'd suggest that we instead understand 'rule by the people' to mean that everyone is able to make a meaningful contribution to the collective decision-making process, over time. The votes of a permanent minority are pointless, as they never have a chance of making a difference. But in a more flexible political culture, "the majority" is constantly in flux. Each person will be in the majority on some issues (and in the minority for others), so their will is at least sometimes heeded. In this sense, they all contribute to the state's decision-making. Even if they do not always get their way, there is still a meaningful sense in which we can describe this as a polity ruled by all (diachronically).
However, it is consistent with such 'diachronic democracy' that everyone be completely dogmatic. At least there are no stable factions, and thus no consistently oppressed (or effectively disenfranchised) subclass of the citizenry. But it would still be the case that for each particular decision, those in the minority were simply disregarded, their "contributions" effectively nullified. The system is effectively a rotating oligarchy, where everyone gets to take a turn.
This raises the question: is synchronic democracy - rule by all at once - possible? I want to suggest that it is possible, so long as the political system is sensitive to and responsive to reasons that any may put forward. In the absence of faction and dogmatism -- better, in the presence of civic respect -- even those who are initially in the minority have a real chance of affecting the outcome, by convincing others of the virtues of their position. Since the outcome is influenced (ideally: determined) by the strength of reasons, and these reasons may be contributed by anyone, it follows that any can make a meaningful contribution. 'Democracy' in the fullest sense is thus realizable in the form of deliberative democracy.