Rule of law and democracy are not at all the same thing. They are probably related -- hard to have meaningful elections without reliable laws about them, for instance. But it's necessary to explain that connection more carefully -- and find out which ways the causality goes -- before you can argue for democracy on the basis of it promoting rule of law.

There are many examples of non-democratic governments that have reliable, predictable, and reasonably even-handed legal process. (Imperial France under Napoleon or Napoleon III had this reputation, as did Rome under the good emperors. Singapore is a modern example.) There are also plausible examples of democracies that don't have reliable legal systems, I suspect.

I'm not sure I would have said that Ancient Athens had "rule of law" as we understand it, for instance.