Sorry about not adding my solution. I tried several times, but I was fighting with the stupid captcha applet.
Anyway, I discovered that there was an old, manually installed Firefox 3.6 in /opt/firefox. This is what the icon was trying to start. I deleted that folder, then the icon wouldn't start anything.
So, I uninstalled and reinstalled FF 4 using aptitude. Then, I found that I could start FF 4 from the terminal, but the icon was still looking for /opt/firefox.
I could not find out how to fix that in Unity, so I logged into Gnome Classic and changed the Properties of both the icon and the menu item for Firefox. Then, when I logged back into Unity, the icon worked.