From the sound of it, the OS doesn't know InDesign is the default application for .indd files and you have an out-of-date link in your start bar. The CC app knows what InDesign is, but the OS doesn't, so the remedy is to either set up the default manually or uninstall and reinstall InDesign. the installation process will add the necessary defaults to the OS.
This happened, possibly, because you uninstalled an earlier version of InDesign after you had installed the new one. This is a frailty of the uninstaller (all of the Adobe apps suffer the same problem) that I hope will be fully addressed soon. At present, the necessary sequence to avoid problems (if you don't want to keep an earlier version of an app) is: a) install new version, b) migrate preferences, etc., c) uninstall old version, d) uninstall, then reinstall new version.
Thanks for the response.I had been worried that it might have been an issue with an older verison of Adobe, so I had the computer rebuilt from scratch so that I wouldn't have that issue. I did find an interesting error in the event logs when I try and use the Shortcut in the start menu. Apparently, the application is attempting to launch, but it is failing.
Faulting application name: Indesign.exe, version: 10.0.0.370. time stamp: 0x5362b4ea
Faulting module name: PMRuntime.dll, version : 126.96.36.199, time stamp: 0x5362b3b6
Exception code: 0xc000000d
Start by updating InDesign to version 10.1.
Thanks. Updated to 10.1.0.70 x64 Build.
I'd try replacing the PMRuntime.dll file with another one, either from a different computer or one installed in an earlier version of ID (as far as I can tell it's the same version going back to at least CS3).