We consider one-dimensional discrete-time random walks (RWs) of $n$ steps, starting from $x_0=0$, with arbitrary symmetric and continuous jump distributions $f(η)$, including the important case of Lévy flights. We study the statistics of the gaps $Δ_{k,n}$ between the $k^\text{th}$ and $(k+1)^\text{th}$ maximum of the set of positions $\{x_1,\ldots,x_n\}$. We obtain an exact analytical expression for the probability distribution $P_{k,n}(Δ)$ valid for any $k$ and $n$, and jump distribution $f(η)$, which we then analyse in the large $n$ limit. For jump distributions whose Fourier transform behaves, for small $q$, as $\hat f (q) \sim 1 - |q|^μ$ with a Lévy index $0< μ\leq 2$, we find that, the distribution becomes stationary in the limit of $n\to \infty$, i.e. $\lim_{n\to \infty} P_{k,n}(Δ)=P_k(Δ)$. We obtain an explicit expression for its first moment $\mathbb{E}[Δ_{k}]$, valid for any $k$ and jump distribution $f(η)$ with $μ>1$, and show that it exhibits a universal algebraic decay $ \mathbb{E}[Δ_{k}]\sim k^{1/μ-1} Γ\left(1-1/μ\right)/π$ for large $k$. Furthermore, for $μ>1$, we show that in the limit of $k\to\infty$ the stationary distribution exhibits a universal scaling form $P_k(Δ) \sim k^{1-1/μ} \mathcal{P}_μ(k^{1-1/μ}Δ)$ which depends only on the Lévy index $μ$, but not on the details of the jump distribution. We compute explicitly the limiting scaling function $\mathcal{P}_μ(x)$ in terms of Mittag-Leffler functions. For $1< μ<2$, we show that, while this scaling function captures the distribution of the typical gaps on the scale $k^{1/μ-1}$, the atypical large gaps are not described by this scaling function since they occur at a larger scale of order $k^{1/μ}$.