Sort of... eventually?
Today, France is one of the world's most prosperous democracies, and it probably wouldn't be if the monarchy had remained in power during that whole period (though it very likely would have become more of a constitutional monarchy like the UK has). In that sense, then, the French Revolution did bring about democracy.
Also, immediately after the French Revolution the government, called the First Republic of France, was relatively democratic, but this government was rapidly overthrown. There were a series of coups d'etat, making it so no government remained in place for more than a few years before the next one took over. The first really stable government after the revolution was Napoleon's, and that was assuredly no democracy — it was basically a totalitarian dictatorship.
For a decade and a half after Napoleon was overthrown, there was the Bourbon Restoration, which was more of a conventional monarchy. That was then overthrown in the July Revolution, which instituted another monarchy. In 1848, another revolution led to the Second Republic of France, which fell a few years later and led to the establishment of another monarchy. Finally, in 1870 — 80 years after the initial revolution — there was one final revolution that established the Third Republic of France, which was a stable democracy for 70 years. The Third Republic held until 1940, when it was conquered by Nazi Germany for a few years, then liberated by the US and UK, and re-established as the Fourth Republic, which was finally reformed (not a coup, for once, though it did emerge in a crisis) into the Fifth Republic we know today.