I found a solution posted in 2013. It worked!
After looking at a new user's registry HKEY_CURRENT_USER and comparing it to the one on the original user, I found a registry value which was not changed, and should have been.
I found a key:
HKEY_CURRENT_USER\Software\Classes\Applications\Photoshop.exe\shell\open\command
This key was set to the path for Photoshop CS6 when it should have been set to Photoshop CC.
In other words, in my case, it was set to:
"C:\Program Files\Adobe\Adobe Photoshop CS6 (64 Bit)\Photoshop.exe" "%1"
When I changed the CS6 to a CC, in other words, so the whole key reads:
"C:\Program Files\Adobe\Adobe Photoshop CC (64 Bit)\Photoshop.exe" "%1"
Everything now works perfectly.
There's no question this is an installer bug and I imagine it's going to hit anyone who upgrades from CS6 to CC. The fix is easy if you are comfortable with editing the registry but it should be fixed in the installer. This is an Adobe bug, plain and simple.
If you create a new user, since it has no keys for HKEY_CURRENT_USER it creates them properly.