This is very clearly a vitally important moment in the ongoing fued between Jack and Ralph. Jack has shown his own abilities in terms of successfully hunting pigs and managing the hunting group of the boys, but now the sightings of the "beast" have given him the excuse he needs to challenge Ralph's leadership publicly and try to gain the position of leader himself. However, having publicly challenged Ralph in such a way, there is clearly no going back, and Jack realises this as the shame and embarrassment of receiving no votes for his proposition hits him. Note his reaction to the deafening silence that follows his proposition:
The silence continued, breathless and heavy and full of shame. Slowly the red drained from Jack's cheeks, then came back with a painful rush. He licked his lips and turned his head at an angle, so that his gaze avoided the embarrassment of linking with another's eye.
His final response of "I'm not going to play any longer," symbolises a irrevocable split between the boys, that Ralph knows can only harm them as they try to co-exist and struggle for survival on this island. Jack recognises that he does not have the support of the boys, and this is a blow from which his pride cannot recover.