Let $\{X_{n}\}_{n\geq0}$ be a sequence of (pointed) spaces such that for all $n \geq 0$, we have homotopy-cartesian squares
$\require{AMScd}$ \begin{CD} X_{n} @>>> \{* \}\\ @VVV @VVV\\ \{ *\} @>>> X_{n+1}. \end{CD}
Does it then follow that $\{X_{n}\}_{n\geq0}$ is a $\Omega$-spectrum?
My thoughts: The homotopy-cartesian squares give you homotopy fibration sequences
$X_{n} \longrightarrow \{* \} \longrightarrow X_{n+1}$,
which imply that we have weak homotopy equivalences
$\Omega X_{n+1} \longrightarrow X_{n}$.
However, to conclude that $\{X_{n}\}$ is a $\Omega$-spectrum, we want weak homotopy equivalences
$X_{n} \longrightarrow \Omega X_{n+1}$.
I'm not sure how (or if) one can obtain such maps?
The question is motivated by the comments after proposition 5.2 of the paper HERMITIAN K-THEORY, DERIVED EQUIVALENCES AND KAROUBI’S FUNDAMENTAL THEOREM, where the author seems to assume this.
Any homotopically commutative square as above (regardless its cartesianess) induces a canonical map $$X_{n}\to \Omega X_{n+1},$$ by the universal property of the homotopy limit (or if you prefer, that of the loop space functor). If the square is homotopy cartesian, this map is the weak homotopy equivalence that you assert. Thus, it goes in the right direction to build a $\Omega$-spectrum.