We perform a Koopman spectral analysis of elementary cellular automata (ECA). By lifting the system dynamics using a one-hot representation of the system state, we derive a matrix representation of the Koopman operator as the transpose of the adjacency matrix of the state-transition network. The Koopman eigenvalues are either zero or on the unit circle in the complex plane, and the associated Koopman eigenfunctions can be explicitly constructed. From the Koopman eigenvalues, we can judge the reversibility, determine the number of connected components in the state-transition network, evaluate the period of asymptotic orbits, and derive the conserved quantities for each system. We numerically calculate the Koopman eigenvalues of all rules of ECA on a one-dimensional lattice of 13 cells with periodic boundary conditions. It is shown that the spectral properties of the Koopman operator reflect Wolfram's classification of ECA.