How would you respond if you were forwarded an email by your secretary that clearly wasn't meant for you?
My first question is what you thought would happen? In my experience, many secretaries would handle it exactly in this way - your email is fairly unspecific in what you thought the secretary would do ("remind him about this" - he/she did by forwarding your mail, and at the same time probably wondering why you did not send it directly to him). If you thought that the secretary would take responsibility for hunting down the recommendation letter for you, I think this is an unreasonable assumption. The secretary is the professor's assistant, not yours. In the olden times this may have included managing the professor's work backlog, but nowadays most people prefer doing this themselves and just having the secretary arrange meetings etc.


  even though it was never written with his eyes in mind.


Tip for the future: never write something to staff where you would be embarrassed if the boss saw it, especially if you don't know the person well. However, I see little in this email that would offend me, other than maybe the implication that you thought I could not be approached directly about this. But of course other people get offended by other things.


  I was wondering how academics would feel if they were put in a similar spot, and if I ought to be proactive in rectifying the situation, or if I should just let sleeping dogs lie.


I am for "letting sleeping dogs lie". Very likely there really isn't a situation to address, and if there is you can still tackle it once it comes up.