This paper studies the estimation of network weights for a class of systems with binary-valued observations. In these systems only quantized observations are available for the network estimation. Furthermore, system states are coupled with observations, and the quantization parts are unknown inherent components, which hinder the design of inputs and quantizers. In order to deal with the temporal dependency of observations and achieve the recursive estimation of network weights, a deterministic objective function is constructed based on the likelihood function by extending the dimension of observations and applying ergodic properties of Markov chains. By imposing an independent Gaussian assumption on disturbances, we show that the function is strictly concave and has a unique maximum identical to the true parameter vector, so in this way the estimation problem is transformed to an optimization problem. A recursive algorithm based on stochastic approximation techniques is proposed to solve this problem, and the strong consistency of the algorithm is established. Our recursive algorithm can be applied to online tasks like real-time decision-making and surveillance for networked systems. This work also provides a new scheme for the identification of systems with quantized observations.