Is it true that homotopy pullbacks and homotopy pushouts coincide in the category of spectra? I had a feeling that this is the case, but don't know where to find a proof or how to prove it. Thanks!

6$\begingroup$ Yes. This is a consequence of the BlakersMassey theorem. $\endgroup$ – Dylan Wilson May 30 '13 at 16:19

14$\begingroup$ if you accept the fact that the category of spectra is stable, then it follows from the axioms of being stable (how tautological was this comment?) $\endgroup$ – Jacob Bell May 30 '13 at 16:21

8$\begingroup$ Dylan, in some sense it's more elementary than BlakersMassey. To get the idea, here's why $X\to \Omega\Sigma X$ is a weak equivalence when $X$ is a spectrum: $X$ consists of spaces $X_n$; the map is given by Freudenthal maps $X_n\to \Omega\Sigma X_n$; an inverse on spectrum homotopy groups is given by the obvious maps $\Omega\Sigma X_n\to \Omega X_{n+1}$. $\endgroup$ – Tom Goodwillie May 30 '13 at 20:32
Although the answer is sketched in the comments, I wanted to remark that this statement is proved carefully by Cary Malkiewich as Proposition 6.2.11 in Parameterized Spectra, A Low Tech Approach. He credits Model Categories of Diagram Spectra as the first place this was proven, but I couldn't find the result there.

3$\begingroup$ First of all, that proof is unreadable :), second of all, he only reduces from the case of parametrized spectra to usual spectra and cites a different paper for that. $\endgroup$ – user147129 Jun 18 '20 at 4:16

$\begingroup$ Still, good to have references for the result that one can cite. $\endgroup$ – David White Jun 18 '20 at 12:56