f you did have any 3rd party extensions installed and did not uninstall them prior to uninstalling 10.0, the Desktop software may not uninstall correctly.
The way we had to fix this in versions past is to reinstall the older software (10.0 in your case?), then, use Control Panel > Programs and Features (as Jayanta pointed out) to uninstall all the 3rd party extensions, then the ArcGIS Desktop should be able to be uninstalled. The install 10.2.2 and reinstall any 3rd party extensions (if they are compatible)
This doesn't see to be as much of an issue for upgrades after 10.2.2, but was with 10.0.x and maybe 10.1.x.