It depends if you're talking about the English perspective or the Norman perspective, and the short answer is from the English perspective, kind of yes, but from the Norman perspective, no, except possibly in a loose, political sense. My answer will be mostly drawn from Chapter 3 ('Normanitas') of 'The English and the Normans: Ethnic Hostility, Assimilation, and Identity 1066 - c. 1220' (Hugh Thomas 2003) and from William of Poitiers 'Gesta Guillelmi' (Deeds of William) (Davis and Chibnall edition, 1998), written shortly after the Norman conquest.

First I'll clarify what 'French' would have actually meant in this period. William of Poitiers, writing in Latin, uses 'Francia' and 'Gallia' (i.e. Gaul) interchangeably to describe the kingdom of France (though sometimes 'Francia' refers only to the royal demesne around Paris), which the duchy of Normandy was a part of. He also once uses 'Gallo' to refer to inhabitants of this kingdom in general, once they were in England. However when it comes to the actual groups of people, it's very clear that Normans (Normanni) are not French (Francigenae / Franci). The French were the Franks, whose power base was the area around Paris, and who as Thomas points out, were actually the Normans' chief rivals. The best way to think about France in this period is that it was the kingdom of the Franks / France, but where most groups in this kingdom did not see themselves as Franks / French, but rather Normans, Bretons, Burgandians, or the various Occitan-speakers of the south etc.

As for the Normans, though they had adopted the language and culture of the area they had settled, the sources (e.g. William of Poitiers, and the writer Dudo who wrote Historia Normannorum around 50 years before the conquest of England) show that Normans had a strong sense of their own identity as Normans, for example by emphasising their heritage and history, and that they were a people 'destined for conquest'. Practically, the duchy of Normandy was an independent realm with its own laws and church, though technically part of the kingdom of France. William of Poitiers describes Normandy as William the conqueror's 'patria' (fatherland), and that he loved Normandy and Normandy loved him! (after the conquest). The Normans are always distinguished from the other groups in the invasion force; one sentence describing part of the battle of Hastings is illuminating: "Pressing home the attack were men of Maine [Cenomanici], Frenchmen [Francigenae], Bretons [Britanni], Aquitanians [Aquitani], above all Normans [Normanni]".

In England though it's a different story, as local distinctions are often lost when closely related peoples find themselves together in foreign countries. Whenever English legal charters distinguish between the English and the Normans etc., the word used for the French groups is generally the equivalent of 'French', not Norman ('Norman' referred to the Norse in general). This is partly because of the multitude of 'French' groups who took part in the Norman conquest, and partly because even before the conquest, Normans and other 'French' who had immigrated to Britain during the reign of Edward the confessor had always been referred to as French. However the English sources from after the conquest are mixed in describing the Normans as Normans or as French, so it may be that with Normans in control of England, their identity came to be more recognised as Norman rather an as generic 'French' (only to merge into English in the subsequent centuries).

Further question: I just saw a youtube clip stating that many of the men involved in the 1066 conquest were direct descendants of Ragnar Lothbrok and his brother/family, as might be expected given why the "Normans" ruled Normandy at all. Are there any things we can point to about their culture and say 'this is a result of that history'?