There is a slight ambiguity what "the ABC conjecture" is as there are some variation. However, the most common and what you likely mean is this fomulation (or something equialent to it):

For every $\epsilon >0$, there is a $C_{\epsilon}$ such that: if $a+b=c$, with positive coprime intergers, then
$$c < C_{\epsilon} \ \text{rad}(abc)^{1 +\epsilon}.$$

Now, for the equation $x^n + y^n = z^n$ this means that
$$
z^n < C_{\epsilon}\ \text{rad}(x^ny^nz^n)^{1 +\epsilon}.
$$
Yet $\text{rad}(x^ny^nz^n) = \text{rad}(xyz)$ so one actually has
$$
z^n < C_{\epsilon} \ \text{rad}(xyz)^{1 +\epsilon}.
$$
and since $z$ is the largest and since $\text{rad}(a) \le a$ this further means
$$
z^n < C_{\epsilon} z^{3 + 3\epsilon}.
$$
Now, take $\epsilon =1/4$, say. Then on the one hand this cannot hold for *any* $z>1$ for $n$ sufficiently large (so no solution for large $n$, what you ask) and also not for any $n \ge 4$ fixed for $z$ sufficiently large (so only finitely many for fixed $n$) or also only finitely many couples $(z,n)$ that fulfill this (for $n \ge 4$).

[Added:] Since the question was changed to remove the gcd condition in Langs's version, I add for completeness, that each solution (for given $n$) implies the existence of a coprime one (for this $n$) so since above establishes there are *no* coprime solutions for some $n$ then there are none at all. [End Added]

However, to make these things effective/explicit one would need to know something about $C_{\epsilon}$ (in dependence of $\epsilon$).

Regarding "strongest possible": I think this is about what can be said, from the conjecture in the way I stated it. If one assumes stronger conjectures where one would have an explicit dependence of $C_{\epsilon}$ on $\epsilon$ then one could give explicit bounds. But (I think) the argument essentially always passes throught the last displayed equation and checking what this yields.