I would say that the French Revolution did little to spread Democracy. The fault of the assumption that it was the Revolution is how history ties the French Revolution to the Napoleonic Wars. For this, I must do some background.

There is no real distinction between the Revolutionary era and the Napoleonic era. As a result, the two eras are folded into a Revolutionary/Napoleonic era. The problem with this is that thematically within the history, the two are different. The Revolution was a time of political madness and indecision as a result of the many players of the Revolution. The Napoleonic Era is defined by the military actions of the Emperor as everything flows from and into his victories.

So, with this I would need to argue that the era of Revolution before Napoleon had very little to do with the spread of Democracy. Political revolutionary ideology was contained mainly in France as the rest of Europe pressed down on any hopes of an international European Revolution. The spread of Democracy has more from the Napoleonic Wars themselves.

Due to the spread of Napoleon's armies, the Nationalist ideology would slowly be spread in everyday European society. Further, in 1813 when most of Europe was United against Napoleon, they pulled on proto Nationalist feelings in order to get the people to work at fighting against Napoleon.

Further, the Napoleonic Code was implemented in many conquered or created lands which had very pro French leadership. I would put more emphasis on the Napoleonic Code (being egalitarian and secular) rather than the ideology of the Revolution itself. Oddly enough, a hereditary Emperor did more for Democracy than an elected Republican government.