I just started having the same problem. I have 3 Gmail accounts and none of them are auto syncing. I can manually refresh and get the emails and I can send . Also, the new mail notifications in the notification bar stopped working at the same time as did the audible notification. Have checked and reset all settings with no luck.
One of my gmail accounts wont synch either. What's going on all of a sudden?
The verizon salesforce tried to tell me I am low on memory - delete my text messages. Do a factory re-set (definately didn't want to do that).
A technician fixed it one day and then shortly thereafter it wasn't working again. Strange, the other gmail account synchs fine.