"French" is adjective or a collective noun, but don't use it as a countable noun.
Trying to say "as a French" makes about as much sense as thinking "as a American" is correct.
As has already been said ... "a French (wo)man","a French person","a French citizen" is the correct way to go.
The reason you can say "an American" is because America starts with a vowel.
Same reason why you would not say "a British" but you could say "a Brit".