Question. Let $\Gamma$ denote a first-order signature, and consider a sentence $\tau$ and a set of sentences $\Sigma$ in the language generated by $\Gamma$. If every countable $\Gamma$-structure that satisfies $\Sigma$ also satisfies $\tau$, do we necessarily have that $\Sigma$ proves $\tau$?
In other words, is first-order logic complete with respect to countable structures?
Almost.
First note that if $M$ is an infinite model of $\Sigma$ then there is some $M'$ which is countable and elementary equivalent to $M$. So if $\tau$ is true for all countable models of $\Sigma$ it is true for all infinite models of $\Sigma$.
But if $\Sigma$ has finite models not satisfying $\tau$, then of course it cannot prove $\tau$.
(Interestingly, if $\tau$ is true in every infinite model of $\Sigma$ then there is some $k$ such that every finite model of size $\geq k$ must satisfy $\tau$.)
That been said, if you consider finite as countable in this context, then the above answer immediately changes to "Yes" for the same reasons. Every model is finite, or elementarily equivalent to a countably infinite model.