This is completely possible. If you place any program (script or binary) on your Desktop and ensure that it has execute permission, it will be executed by double-click by the vast majority of desktop rendering software (very definitely thunar and nautilus, but anything else that follows the same specifications). Note that some desktop handlers may ask for confirmation that you wish to run, rather than view, the program, depending on the implementation of the program (this is in part to workaround a historical issue that files from FAT tend to all be marked executable, even when one doesn't want to execute them).
If you prefer a pretty icon, create a .desktop file on your desktop referencing the program you wish to run. If this is granted execute permission, any compliant desktop rendering program will launch the program as described in the .desktop file, and should do so without asking for confirmation in any way.
The specification for creation of .desktop files is http://standards.freedesktop.org/desktop-entry-spec/latest/ , or you might start with something already created under /usr/share/applications/ . You can check to make sure your .desktop file is correctly written with the 'desktop-file-validate' program from desktop-file-utils.