The format cannot be the problem because iMovie only exports in Mp4 or Pro Res, both of which are supported formats. Something else is wrong.
To cover the bases, is it possible that you are trying to open the temporary file that gets deposited on the desktop while an export is taking place? If so, it could not be opened or played. The file will disappear shortly after the export is complete and the full sized Mp4 file will then appear in its place. Do a Control-click/Get Info on the file you are trying to open and see if there is any size content to it. If it says 0 bytes, or the file size is very tiny, then the export failed.
Try deleting/resetting preferences and see if that cures the problem. Open iMovie while holding down the Option and Command keys and select to delete preferences in the box that appears. iMovie will open in a new library. Reopen your old library to get back to your projects. Now see if you can share a playable movie to your desktop. Possibly you may need to reboot to set the changes. Deleting preferences is a safe procedure that will not cause data loss or disruption to your project.
-- Rich