Let us start from some unitary $U$ with corresponding spectral decomposition $U=\sum_{\lambda\in\sigma(U)}\lambda P_\lambda$. Here, $\sigma(U)$ denotes the spectrum (i.e. set of eigenvalues) of $U$ and $P_\lambda$ are the spectral projections, that is, $P_\lambda^\dagger=P_\lambda$ and $P_\lambda P_{\lambda'}=\delta_{\lambda,\lambda'}P_\lambda$ for all $\lambda,\lambda'\in\sigma(U)$, as well as $\sum_{\lambda\in\sigma(U)}P_\lambda={\bf1}$.
To put this into a more familiar quantum info context the spectral projections are of the form $P_\lambda=|g_\lambda\rangle\langle g_\lambda|$ or, if $\lambda$ has multiplicity larger than one, $P_\lambda=\sum_k|g_\lambda^{(k)}\rangle\langle g_\lambda^{(k)}|$.
In any case the corresponding channel induced by the gate $U$ can be written as
$$U(\cdot)U^\dagger=\sum_{\lambda,\lambda'\in\sigma(U)}\lambda(\lambda')^*P_\lambda(\cdot)P_{\lambda'}\,.\tag1$$
On the other hand $U$ (more precisely: the Hamiltonian $H$ which generates $U$ via $U=e^{iH}$) induces a projective measurement through its spectral decomposition. Thus if some outcome ($\log\lambda$) is measured on some system in the initial state $\rho$, then the corresponding post-measurement state is
$$
\frac{P_\lambda\rho P_\lambda}{{\rm tr}(\rho P_\lambda)}\,.
$$
Thus these formalisms are connected by the fact both of them are determined by the spectral decomposition—and, in particular, the spectral projections $P_\lambda$—of the unitary $U$.
In addition, there is a scenario where a measurement can be expressed as a quantum channel (just like the gate transformation $\rho\mapsto U\rho U^\dagger$ is a quantum channel).
To every measurement one can associate a "expected density operator" which can be interpreted as the state of the system after measurement if the experimenter does not have access to the measurement outcome. For a projection-valued measurement this channel takes the form
$
\rho\mapsto\sum_{\lambda\in\sigma(U)}P_\lambda\rho P_\lambda\,.
$
In some sense these are then the "diagonal terms" of the channel $U(\cdot)U^\dagger$, i.e. only those terms in (1) for which $\lambda=\lambda'$.
For more on measurement, (expected) post-measurement, and the like, cf., e.g., this other answer of mine or Chapter 3.3 in "Principles of Quantum Communication Theory: A Modern Approach" by Khatri & Wilde.
