Dissipative Kerr solitons arising from parametric gain in ring microresonators are usually described within a classical mean-field framework. Here, we develop a quantum-mechanical model of dissipative Kerr solitons in terms of the Lindblad master equation and study the model via the truncated Wigner method, which accounts for quantum effects to leading order. We show that, within this open quantum system framework, the soliton experiences a finite coherence time due to quantum fluctuations originating from losses. Reading the results in terms of the theory of open quantum systems allows us to estimate the Liouvillian spectrum of the system. It is characterized by a set of eigenvalues with a finite imaginary part and a vanishing real part in the limit of vanishing quantum fluctuations. This feature shows that dissipative Kerr solitons are a specific class of dissipative time crystals.