First let me thank Urs, Karol, and Rune Haugseng for helpful comments.

Now note that the inclusion, $i$, of $\infty$-groupoids into $(\infty,n)$-categories has an $\infty$-categorical left adjoint, $L$ (for lack of a better name), and a right adjoint $(-)^\prime$.

Given a pointed $(\infty,n)$-category $\mathcal{C}$, the loop category $\Omega \mathcal{C}$ is defined by the following (homotopy) pullback diagram:

\begin{array}{ccc}
\Omega\mathcal{C} & \rightarrow & \ast\\
\downarrow & & \downarrow \\
\ast & \rightarrow & \mathcal{C}
\end{array}

Now $\mathcal{C}^{\prime}$ is the maximal sub-$\infty$-groupoid of $\mathcal{C}$ (the core). Since $\ast$ is an $\infty$-groupoid, the inclusion $\ast\rightarrow \mathcal{C}$ factors canonically through $\mathcal{C}^\prime$. By a standard finality argument we see that $\Omega \mathcal{C}$ is equivalent to the homotopy pullback:
\begin{array}{ccc}
\Omega\mathcal{C} \simeq \Omega^{Top}\mathcal{C}^\prime& \rightarrow & \ast\\
\downarrow & & \downarrow \\
\ast & \rightarrow & \mathcal{C}^\prime
\end{array}

Regarding $\mathcal{C}^\prime$ as a space (since it is an $\infty$-groupoid), we see that $\Omega\mathcal{C}$ is equivalent to the space of topological (based) loops on $\mathcal{C}^\prime$ since $i$ preserves (homotopy) limits.

Unraveling this a bit, we see that $\Omega(-)$ is naturally equivalent to $i\Omega^{Top}(-)^\prime$ which is a composite of right adjoints. It follows that the left adjoint, $\Sigma(-)$, is naturally equivalent to $i\Sigma^{Top}L(-)$.

As a consequence, an $(\infty,n)$-category which is a loop category is necessarily a loop space. This shows that the inclusion of spectra objects in $(\infty,0)$-categories (i.e., spectra) into spectra objects in $(\infty,n)$-categories is an equivalence. So the two categories have the same stailizations.