Stabilization of $\infty$-categories versus SW stabilization Spanier-Whitehead stabilization provides a way to extend a category $\bf E$ to a bigger one $\mathcal{SW}_\Omega(\bf E)$ where a given endofunctor $\Omega$ is invertible. The category $\mathcal{SW}_\Omega(\bf E)$ is constructed with


*

*Objects the pairs $(A,n)\in Ob(\mathbf C)\times\mathbb Z$;

*The set of morphisms $(A,n)\to (B,m)$ corresponds to the colimit set
$$
 \varinjlim_{k\in\mathbb N} \hom_{\bf E}(\Omega^{n+k}A, \Omega^{m+k}B)
$$


It's a matter of bare computations to show that it defines a category, where $\bf E$ can be embedded via $A\mapsto (A,0)$, and where an entire family of functors $\bar\Omega^i\colon (A,n)\mapsto (A,n+1)$ can be defined; the functor $\bar\Omega^1$ plays the role of the initial endofunctor $\Omega$, and that's the end of the story.
But when you meet the formalism of stable $\infty$-categories you begin to wonder if there's a link between the two processes, the SW stabilization and what Lurie describes here  (Def. 8.4). I'm aware that the SW construction is an "abstraction" (?) of the procedure exhibiting topological spectra, but I must confess I'm not able to go further (especially because I'm a "category theorist" slightly oriented to topology, not vice-versa).
 A: I think it's more typical to use this construction to invert the suspension functor, rather than the loop functor. So let me write $\Sigma$ where you wrote $\Omega$.
1) The category $SW_{\Sigma}(\mathcal{C})$ can be identified with the direct limit of the sequence
$$ \cdots \rightarrow \mathcal{C} \stackrel{\Sigma}{\rightarrow} \mathcal{C} \stackrel{\Sigma}{\rightarrow} \mathcal{C} \stackrel{\Sigma}{\rightarrow} \cdots$$
2) The construction makes sense for $\infty$-categories as well as ordinary categories. Moreover, it commutes with passage to homotopy categories. That is, if $\mathcal{C}$ is an $\infty$-category, then the homotopy category of $SW_{\Sigma}(\mathcal{C})$ is $SW_{\Sigma}( h \mathcal{C} )$. 
3) Let $\mathcal{C}$ be a small pointed $\infty$-category which admits finite colimits, and let $\Sigma: \mathcal{C} \rightarrow \mathcal{C}$ be the suspension functor. Then $Ind( SW_{\Sigma}( \mathcal{C} ) )$ is the stabilization of $Ind( \mathcal{C} )$. 
You'd typically put these together by taking $\mathcal{C}$ to be something like the $\infty$-category of pointed finite spaces. Then $Ind( \mathcal{C} )$ is the $\infty$-category of pointed spaces, and its stabilization is the $\infty$-category of spectra. Your conclusion is that $SW_{\Sigma}(\mathcal{C} )$ can be identified with the $\infty$-category of finite spectra, so that
$SW_{\Sigma}( h \mathcal{C} )$ is a model for the homotopy category of finite spectra (this is the classical construction).
