Let $f:\mathbb{R} \to \mathbb{R}$ be a stationary centered Gaussian process. For any $R>0$, let $ν_R$ denote the counting measure of $\{x \in \mathbb{R} \mid f(Rx)=0\}$. In this paper, we study the large $R$ asymptotic distribution of $ν_R$. Under suitable assumptions on the regularity of $f$ and the decay of its correlation function at infinity, we derive the asymptotics as $R \to +\infty$ of the central moments of the linear statistics of $ν_R$. In particular, we derive an asymptotics of order $R^\frac{p}{2}$ for the $p$-th central moment of the number of zeros of $f$ in $[0,R]$. As an application, we derive a functional Law of Large Numbers and a functional Central Limit Theorem for the random measures~$ν_R$. More precisely, after a proper rescaling, $ν_R$ converges almost surely towards the Lebesgue measure in weak-$*$ sense. Moreover, the fluctuation of $ν_R$ around its mean converges in distribution towards the standard Gaussian White Noise. The proof of our moments estimates relies on a careful study of the $k$-point function of the zero point process of~$f$, for any $k \geq 2$. Our analysis yields two results of independent interest. First, we derive an equivalent of this $k$-point function near any point of the large diagonal in~$\mathbb{R}^k$, thus quantifying the short-range repulsion between zeros of $f$. Second, we prove a clustering property which quantifies the long-range decorrelation between zeros of $f$.