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We consider a joint uplink and downlink scheduling problem of a fully distributed wireless networked control system (WNCS) with a limited number of frequency channels. Using elements of stochastic systems theory, we derive a sufficient stability condition of the WNCS, which is stated in terms of both the control and communication system parameters. Once the condition is satisfied, there exists a stationary and deterministic scheduling policy that can stabilize all plants of the WNCS. By analyzing and representing the per-step cost function of the WNCS in terms of a finite-length countable vector state, we formulate the optimal transmission scheduling problem into a Markov decision process and develop a deep reinforcement learning (DRL) based framework for solving it. To tackle the challenges of a large action space in DRL, we propose novel action space reduction and action embedding methods for the DRL framework that can be applied to various algorithms, including Deep Q-Network (DQN), Deep Deterministic Policy Gradient (DDPG), and Twin Delayed Deep Deterministic Policy Gradient (TD3). Numerical results show that the proposed algorithm significantly outperforms benchmark policies.