I got the impression from WCog's answer that patient would not be charged because the doc opted not to respond in any clinically meaningful way, that is: "can't help you by email on this..."
Regarding attorneys, while they do tend to 'have it made in the shade', I don't think I want to start thinking or charging like an attorney. That's just me.
For one thing, before I open the email can 'o worms, I'd like some precedents. What if you don't answer in a timely fashion?
Patient: I'm having chest pain, what should I do?
Say you don't get to that one? I dunno...it goes to junk mail, or you just miss it or something, then what?
And you mistype something or forget a disclaimer and POW!!! your email correspondence is available to testify against it's creator--you!
I DO do some email with a paltry few who I happen to know well on a personal level outside the office, but not much else.