Yes, that very much effectively makes him a Frenchman.

More to the point, Napoleon adopted the French nationality; as a young Officer he was passionately and deeply involved with the idea of Corsican independence, but his relatively poor interactions with leaders of that movement eventually alienated him. By the time he was a commissioned Captain, the young Bonaparte had fully attached himself to the idea of France and a French identity; shortly after the Siege of Toulon he begun to sign his name as 'Bonaparte' in the French style, opposed to 'Bunoparte' or even earlier 'di Bunoparte.'

Sources: The Campaigns of Napoleon, David Chandler + "Napoleon", Alan Schom.

Napoleon might have considered himself French, but what he considered "French" was not exactly what his contemporaries thought was indicative of being French.

The whole Bonaparte family was caught up in Corsica's transfer to French control and this complicated their sense of national identity. Napoleon's father, Carlo, was an assistant to the Corsican nationalist Paoli, but very quickly accommodated himself to French rule. Carlo subsequently became a notorious social climber and tried to establish that the Bonapartes were not just the minor nobility they became when the French took over. Both Napoleon and his brothers became agents of Carlo's attempts to cement their credentials as French nobles and enhance the family's position.

Napoleon's hemming and hawing between a French and Corsican identity before he elected to self-identify as French has to be seen in this context. In military school, he adapted the attitude of a Corsican patriot even as he learned French customs and mores. But the young Napoleon also had an appreciation for the role of networking and operating within existing patronage systems that characterized the ancien regime. He had witnessed and been a part to his father's schemes to elevate the Bonaparte clan above their relatively humble station. And to a certain extent, Carlo's plan worked. His children, and not just Napoleon, managed to navigate the chaos of the Revolution very well despite the loss of their Corsican estate in 1785. This inculcated a very functional approach to identity within the young Napoleon which he then would apply to his style of imperial rule. Napoleon's Civil Code favored the concept of jus soli (citizenship based upon land of birth) over jus sanguinis (citizenship based upon parent's nationality). This was something that Napoleon had to fight against some elements within the French imperial government when he pushed for jus soli. Napoleon believed that by creating French-led cultural and state institutions, he could assimilate large sections of Europe into French culture and they would (and probably more importantly in Napoleon's calculations) discharge their responsibilities to the nation by paying taxes and serving in the army.

The best source for Napoleon's early life (way better than Schom) is Dorthy Carrington's Napoleon and His Parents: on the Threshold of History which details how the whole clan, not just their illustrious son, had to navigate the strange liminal space between Italian and French identity.