You are true indeed thinking that "Our ancestors the Gauls" is a 19th century idea.

During this century countries are creating some national myths with heroes and symbols. France was looking for a reference that would be unrelated to previous kings. The kings of France had largely used their so-called connections to the Francs and Clovis while in search of legitimacy. To compete with these, the Republic of France needed an older figure and the Gauls proved to serve the purpose well.

Let's be honest, very little was known from the Gauls back in the days, except from what you could gather from the Belli Gallicum written by Caesar. It was then really easy to create something out of it. The myth itself became highly propagated with the Republican free schooling. With the laws Jules Ferry passed in 1881-1882, school became free and mandatory for all kids under 13. The Republic never hid its intention to make this a good way to produce good citizens (hear the political assumption of the word here), hence it needed and created some national heroes and myths.

In the same idea were also "created" Jeanne d'Arc, Clovis, the lazy kings ("les rois-fénéants") and all sorts of funny little depictions of the kings and knights of old.

On an ironical note, as the French colonial empire was wide but the Republic wanted its schooling as equal as possible, this teaching was spread in all the colonies and I, myself, have met a guy from Libreville, in Gabon, who was taught History with that reference to his "Ancestors the Gauls" while he was descending from Pygmeans. Those myths put in place in the 19th century are still vivid and it is still taught more or less that way nowadays in France.

Regarding the second part of your question, to be highly radical I'd say there's no less accurate statement than saying French culture is related to the Gauls. Gauls were an entity of many tribes, part of a wider population of celts spread, to put it loosely, on all Western Europe (As this is vastly dependant on the most recent findings, this notion is evolving and shifting quite quickly nowadays). The limits are still vague between Gauls and Celts as they didn't have a proper written language of their own and really rarely wrote down things; we think their culture was widely based on oral communication. In terms of History that means the Gauls are part of proto-history (or at least we call it that way in France) and we don't know much of this culture. The only knowledge we have, apart from cunning re-readings of the Galli Bellicum, is what Archaeology managed to dig from the ground and we didn't have that much information.

Some linguists have analysed French language and could say we have about 200 words coming from Gaul languages, mostly toponyms. This little indicator shows how there is no to few influences on a cultural level from the Gauls in French culture nowadays.

If you want to go further into this, I only have French references unfortunately (Pardon my French :p): "Nos ancêtres les Gaulois et autres fadaises" from Francois Reynaert, killing all those old French myths in History "Les gaulois du midi" from Michel Py, really proficient archaeologist regarding Gauls; the book is a bit old now but still accurate for the bigger part

I can't speak of specifically the French situation, but I have put some research into the rise of nationalism and the representation of Roman history (Tacitus and Caesar) in Belgian and Dutch historiography of the nineteenth century.

Bluntly speaking all claims of direct influence on modern culture of these 'barbaric' tribes can be written off as nonsense. There has been so much migration of tribes in the early Middle ages that it is highly doubtful that the tribes that were described by Tacitus remained in the same location. It is even less likely that elements of that culture remained in place (excepting maybe those that were tied to the geography rather then the particular tribe, e.g.. People living in the area the Cananefates lived in would be fishermen and seafarers due to the abundance of water).

That being said: when people started looking at national traits and the origin of their people in the 19th century they started extrapolating wildly from these kinds of sources. The Batavians became the mold of the Dutch people, because they had revolted against the Romans under Julius Civilis (note the Latinized name, meaning him or his ancestors were made a Roman citizen, more on that later). This revolt against foreign oppression rhymed beautifully with the revolt against Spain that made the Dutch Republic independent and was therefore used as a Dutch myth of origin.

Note the term 'mythical'. These concepts were heavily politicized. For instance, after 1830, when Belgium became independent from the Dutch kingdom you see that in Dutch history books (certainly the less academically acclaimed ones) the deceitful Belgian tribes get the blame for the failure of Civilis' revolt. This was considered to be because the essentially Gallic cultured Belgians didn't have the resolve of the more Germanic Batavians.

Of course, like I stated, those are all myths, and these myths are political tools in creating a national essence of the people of a country. In reality the tribes all Romanized due to exposure to Roman culture and many Roman soldiers settling down in the region (see Civilis' Latin name here). No actual 'national essence' could have persevered through the ages as there was no nations to speak of: people had a more local sense of belonging tied to their direct surroundings. This would stay that way in most countries until the 19th century.

Despite that, Tacitus' writings have had massive impact on how people thought on the subject of national essence. Christopher Krebs describes in his book 'A most dangerous book' how national historic essentialism shaped, for instance, the ideas of the Nazi's about the Master Race. Also read 'The invention of tradition' by Hobsbawm on how the 19th century shaped our ideas about national traditions.

Now, to go back to France and Sarkozy's mention of the Gauls: that has probably to do with him trying to resort to the notion of a certain national essentialism of the French. This is noteworthy because the French didn't go as far in this kind of cultural determinism, instead thinking that anyone could, by way of civilization, learn to be French. This was different from the German idea that one could only be born with true German culture. But like I said, I can't speak too much on this because it is not really my field, I will gladly yield the floor to a specialist in French history here.

The link to the Gauls is something drummed up by the French Republic during he 19th century because it would conveniently make the Catholic Church disappear.

So is "Our ancestors the Gauls" a valid claim? Culturally, not at all. When the Romans annexed Gaul, they pretty much erased any trace of Gaulish culture. The Gauls basically morphed into Gaullo-Roman.

How about demographically? Are the French people now descendant from the Guals? Well, that's something much more valid. First, you have to realize that people didn't use to move a lot. In France, if you can trace your ancestry back to a village (France population was at least 50% rural until the 1930s) at the beginning of the 19th century, you'll most likely trace back the tree there until the beginning. The bicycle is the invention that allowed people to find mates in the nearby village. Before that there wasn't much movement overall.

It is a fancy idea to think the territory of France has seen its population constantly changed by a series of invasions. Overall the population has been much more stable that people seem to say nowadays. The Roman invasion for example: roman colonists were a tiny minority, compared to the 4 or 5 millions inhabitants of Gaul, even in the Narbonnaise, so the overall population didn't change. What they change however is their culture.

The German invasions after 406 were also not so impactful on the population, or it was mostly about destruction. Many Germanic people actually just crossed the country (like the Vandals who went up to North Africa, or the Wisigoth who settled in Spain). The Franks were a tiny people as well, so again, we are talking about tens of thousands vs. several millions.

A great example is the Viking invasions: those who stayed were single men who married local females. The Vikings left almost nothing - culturally speaking - they totally assimilated to the general local population, even though they took the high positions in their areas. They left a few words (mostly related to ships), and a higher proportion of blonds in some parts of Normandy.

So basically, biologically speaking, French people are indeed descendants of the Gauls, more than anything else. That is if they have great-grand-parents originally from Metropolitan France.

But the question is actually, who were the Gauls? Well, they were also invaders too...

If you are interested in that, you can check in French the books from Jacques Dupaquier.

I am not aware of detailed genetic studies, but I found that: http://www.eupedia.com/europe/projet_adn_benelux_france.shtml

As you can see in the map of France, the Celtic (ie Gaul) element, is strong...