I am using Gedit 2.28.3
From the filesystem, I would like to stop the program from showing the "There is a new version of gedit (x.x.x)"
Is there some way to do this?
Well, this is not really a programming question, but I will try to give a somewhat programming-related answer...
The most accurate documentation of a piece of software is its source code; I downloaded the source code for your version of gedit from here and did a search for the string "There is a new version". I got a hit in plugins/checkupdate/gedit-check-update-plugin.c.
It is reasonable to assume that this functionality is provided in a modular manner through a plugin - one that at least my Linux distribution disables explicitly with the --disable-updater configure option when compiling. Since it's a plugin, the easiest way to deal with this would be via the Edit->Preferences->Plugins tab.
If you want to disable the plugin without going through the gedit user interface, you can try using the gconf-editor tool to modify the GConf settings directly. Just navigate to apps/gedit-2/plugins and remove the checkupdate plugin from the list.
If you do not want to use a graphical utility, you might be a ble to modify the underlying GConf file directly - mine is at ~/.gconf/apps/gedit-2/plugins/%gconf.xml. It's an XML file and the format is self-evident...
Disclaimer: modifying GConf files directly is not recommended - you're on your own...
PS: Oh, removing the plugin file, libcheckupdate.so or something similar, might also work - or it might set your house on fire...