I am surprised that your own instinct didn't stop you from sending out that email. One can just imagine how embarrassed your manager much have been when s/he saw the message. Nobody on that email would have been impressed by how you handled it including your manager’s manager. Take what 一千零六夜said above to heart; try to understand the general approach in handling similar kind of issues at work. When in doubt, post a msg here first before taking any action.
You might want to do some damage control by having a chat with the manager and letting him/her know you were too focused on the work that needed to be done and might have not handled the email in the best way, and you’ve learn a good lesson from this.