A novel cooperative automatic repeat request (ARQ) method with network coding is proposed for broadcast networks where single source node broadcasts multiple packets to multiple destinations. Upon a failed transmission, multiple relays cooperate with the source node at the retransmission phase, where network coding is used to combine multiple lost packets. A random access mechanism is employed where retransmission is done according to back-off counters. The proposed method employs the back-off mechanism to achieve space diversity by punishing the nodes which fail at previous retransmissions, and to reduce the number of retransmissions by giving priority to nodes having opportunity to employ network coding. The proposed method is compared with network coded ARQ without cooperation and traditional ARQ using Monte Carlo simulations for different channel settings. The effect of channel correlation is investigated where correlation is modeled as a finite-state Markov model. Simulation results show that the proposed use of cooperation and network coding for ARQ improves throughput performance when relay channels' conditions are better than the channels between source and destinations. It is also observed that, the proposed strategy exploits cooperation and space diversity to reduce the performance degradation due to correlation.