Excision in the stable homotopy category

The only reasonable notion of stable homotopy equivalence of pairs $(A,B) \rightarrow (C,D)$ that is compatible with your question is an equivalence $\Sigma^\infty (A \cup \operatorname{cone} (B)) \rightarrow \Sigma^\infty (C \cup \operatorname{cone}(D))$.

Now given a map of pairs $(A,B) \rightarrow (C,D)$ that induces an isomorphism on relative homology, it is easy to show (provided you know basic stable homotopy theory) that the induced map $\Sigma^\infty (A \cup \operatorname{cone} (B)) \rightarrow \Sigma^\infty (C \cup \operatorname{cone}(D))$ is a stable equivalence. This is because relative homology coincides with homology of the mapping cone of the inclusion (this is a consequence of excision), and a map of finite spectra is an equivalence, if and only if, it is a homology equivalence (this is Whitehead's theorem in spectra).

Hence, the homological statement of excision implies the stable homotopy version of excision, i.e. $\Sigma^\infty (X - B \cup \operatorname{cone} (A-B)) \rightarrow \Sigma^\infty (X \cup \operatorname{cone}(A))$ is an equivalence.