The error 1603 Windows Installer is something that does not help at all. It can be absolutely anything. Then check for a solution to this problem will lead to many different solutions and most of them will not work.
But it worked for me eventually uninstall the previous Skype with Revo Uninstaller Freeware and reinstalling Skype from the offline installer. Yes, must be the offline installer, since the standard installer does not work.
The offline installer is half hidden, HE knows-because. To download it, used that link here:
The problem is recurrent. Happen again every time you need to install an update, then these steps must always be repeated. And I think it will be so until Windows has to be reinstalled.