No, the statement $\forall x \in X : P(x)$ does not require that there be an $x \in X$ (at least, under the standard interpretation). Generally, $\forall$ is taken to mean the same as $\neg\exists\neg$ - in other words, $\forall x \in X : P(x)$ should be the same as $\neg\exists x \in X : \neg P(x)$. In order for this new sentence to be false, we would need to have an $x \in X$ with $\neg P(x)$; for $X = \emptyset$, there is no such $x$, so $\neg\exists x \in X : \neg P(x)$ can't be false, so $\forall x \in X : P(x)$ must be true.
Another way of thinking about it is this: if I say "all unicorns are blue", the only way to prove me wrong is to supply a unicorn which is not blue - I'm not claiming that there is a unicorn at all, just that any that do exist are blue. Since there are no unicorns, you can't provide a counterexample; so my statement was true.