Technically yes the demonym is "French", but "I'm a French" just doesn't work in English. The word 'French' is almost exclusively used in English as an adjective or the name of the language. It is never used as a noun for anything else. So in context, it reads as an adjective without a paired noun.

In English, you have to disambiguate be adding a noun: French person, French citizen, or Frenchman if you're old and inconsiderate.

Similarly, we don't call people "a Chinese". That construction is considered derogatory, if not outright racist. Demonyms typically cannot be used as nouns alone without a suffix. "A Brazilian" or "a Spaniard" are acceptable.

As usual for English, the rules are vague and inconsistent.