It's generally believed that Fermat thought he had a proof, but probably almost immediately remembered that not everything is a Unique Factorization Domain, so the "obvious proof" fails. Then he didn't bother returning to correct the error.

So no, probably not.

(+) I should go and learn more about the specifics of this to make sure I'm relating it correctly.

EDIT: (++) OK, here's what I was thinking about:

https://math.stackexchange.com/questions/953462/what-was-lam...

EDIT2: (++) Second link with similar details:

https://math.stackexchange.com/questions/324740/fermats-proo...

I just recently learned that the note in Fermat's margin was published posthumously by his son! So Fermat never necessarily publicly claimed to have a proof. So I would imagine you're absolutely correct.