I didn't want to answer this because the question seemed too elementary to spend time on. But to see quasicategories invoked for something so classically elementary is truly painful. (Forgive me Dylan). Jon asked "to see that spelled out (as if I were a baby)". Here goes. The question was about spectra, and you place yourself in any halfway reasonable category of such. If you are really old-fashioned your objects are CW spectra (alias cofibrant), but to have a homotopically meaningful construction we may as well use CW approximation or model theoretic cofibrant approximation to assume that $A$, $B$, and $C$ are CW or cofibrant. We have standard cylinders $A\wedge I_+$ and form the double mapping cylinder (explicit homotopy pushout) $$D = B \cup (A\wedge I_+) \cup C$$ with cofibration (yes, I know, that requires just a smidgen of trivial verbiage) $B\vee C \longrightarrow D$ with quotient (yes, there is no problem with quotients) $D\longrightarrow \Sigma A$. You are in a stable situation so the cofiber sequence extends to the left, but you can also just look at its extension $\Sigma A\longrightarrow \Sigma B\vee \Sigma C$ to the right to identify the maps. This is all exactly as if you were just in based spaces, except that in a stable situation cofiber sequences are fiber sequences. Nothing at all fancy, and the argument adapts very easily to any stable model category.
Dylan, you asked about classical excision; one way that goes is to use CW approximation to replace an excisive triad $(X;A,B)$ by a CW triad; then, with $C=A\cap B$, $X/C \cong A/C\vee B/C$. (Concise, page 110).