I would love to see them try to do that, especially when there is a doctor shortage in this country. This is my e-mail account, not theirs. Besides, this is how everyone else does it.
Isn't that a HIPAA violation to send out potentially patient information from an encrypted and secure email out to a "free" service like GMAIL so that you can then access it on your mobile device?
I work in a benefits/financial services company and I can tell you I would be fired immediately if I pulled something like that sending company emails, personnel data, etc out of our servers onto a free account just so I can access it on my mobile device.
I know our security and auditors (whether internal or outside) would have a field day with that. I can imagine even JCAHO would not look kindly on a hospital who allows their doctors to send potentially patient specific information via the method you are currently using.
Heck, as a potential patient, I wouldn't want my doctor to be sending emails (potentially about me) to his gmail so he can review it on his iPhone, etc.