Is there an app that lets me use an iPhone as a mouse? I'm looking for an app which enables me to physically maneuver my iPhone on the desk to control my Mac's cursor. (I am not looking for a "remote trackpad" app—there are dozens of those.) 

I did a great deal of searching and found nothing. Is this even possible? If so, is there an app in existence which provides this functionality?
To the best of my knowledge, the answer is no.  Of course, without trying out every single mouse app in the App Store, I can't be 100% certain.

I can tell you that I've been a long time user of Mobile Mouse.  They have about the best mouse/remote control app I've found.  They put a lot of effort into trying to get the app to work like a real mouse, and eventually decided that it couldn't be done, at least not with an acceptable level of performance (see FAQ #6).  They decided on the "air mouse" mode instead, and though it doesn't work like a real mouse, it does work very well for what it is.

Best of luck in your search, and of course if you do find such an app, please let us know!