Like most modern people, the French are the product of something called
ethnogenesis. That is an historical process in which a new ethnic group comes into existence from the blending of a number of ethnic components. Right now, I'm hard-pressed to think of one current ethnic group that has not emerged from ethnogenesis at some point in history. Some are older, some are younger, and some are more extreme than others, but virtually all peoples I can think of derive from others. That is something nationalists have a problem with, because they believe in the genetic identity of ethnicity. They will accept ethnogenesis only in the context of an ethnolinguistic family, such as Germanics, Romanics, and so forth - at the most they will accept it within a greater family such as the Indo-Europeans or the Turkic peoples. The results of this ideology tend to be
cataclysmic.
With the French, the process of ethnogenesis is unusually well documented, because it occurred in historical times, i.e. in times where the concerning region was the subject of historiography and documentation. My Celtic history is a little hazy, but I seem to remember the Celts, or Gauls, dominating what is now France starting with the mid-Hallstatt period, which should be somewhere around 600 to 500 BCE. It is well attested that there was a pre-Celtic population in the region, and I wouldn't call it a stretch to say that the Celts mixed with them. Rather the opposite. Just how big the Celtic group that came into France was, and how the process of Celtic domination took place, is unknown. I seem to remember reading somewhere that what little is known of the language of Celtic Gaul does include some non-Celtic substrates, but again, hazy and all.
So whatever the Romans under Caesar conquered was a country that was inhabited by a predominantly Celtic population, with some other elements at the frontiers, most notably Germanic in the East, as even Caesar attests. I seem to remember that the Aquitanians were also not Celtic. The Romans controlled the area for 500 years, and the Gaulic population became profoundly Romanised, adopting Roman language and culture. Just how much of the Celtic heritage was still in existence when the Franks came in the mid-5th century CE I daren't say. I doubt that, with the continued existence of Celtic gods in the Roman pantheon, the Celts had completely vanished, but this is something I never really looked into. However, what the Germanic Franks found in ca. 450 CE was a population that is historically known to have been Romanised Celts. The Franks were admittedly one of the larger peoples of the migration era, but I find it a bit of a stretch to believe that they replaced the entire Gallo-Roman population, as the French like to put it nowadays. They blended. There are Franks outside of France nowadays, and they are significantly different from the French. Much of western Germany is made up of a population that derives itself from the Franks. They are known in English as Franconians, but call themselves Franks. It is a historical fact that they originate from the Franks of Pippin and Charlemagne, just as the French do. However, they are significantly different in language and culture, because they blended with different populations.
The French themselves long acknowledged their Frankish heritage. Then they started being at odds with Germany, and found the idea of having Germanic origins to be appalling, and started constructing a
Gallo-Romain heritage which they cling to until today. That doesn't mean they completely neglect the Frankish element, but they overemphasise the Gallo-Roman one. As it is, it is propaganda, not history.
An interesting side note: The words Celt and Gaul are the same thing. Celtic is the Greek version of the word, Gaulic the Latin one. Many Celtic tribes have retained that original name, in words like Gaelic, Galicia, Galatia, and so on.
tl;dr: You can't say the French were Celts, and you can't say they weren't.
