Gnome update manager panel icon stuck on indicating security downloads available

There must be some error in your packages, so:

To fully check and upgrade your system using terminal using command line,
open a Terminal from the menu Applications->Accessories->Terminal

Tip: right click with mouse on the terminal title caption and select the item "Always on Top" doing this you will force the terminal window to stay on top of the other windows and you will find very easy to copy single row from this web page into the terminal...
Something more about using the terminal https://help.ubuntu.com/community/UsingTheTerminal

Then type or better copy and paste a row a time then press enter:
(Tip: select the single row to copy then right click into the terminal and to quick paste click with middle button of mouse )

sudo dpkg --configure -a
sudo apt-get -f install
sudo apt-get --fix-missing install
sudo apt-get update
sudo apt-get dist-upgrade
sudo apt-get clean
sudo apt-get autoremove
sudo apt-get update

give your user password when requested, you don't see nothing when you type it, then press enter.

Hope this helps

Thank you! Those commands updated and cleaned up a whole slew of packages, but the Update Manager panel icon is still indicating that there are security downloads available, even though after opening Update Manager indicates that all my packages are up-to-date.

Try reboot.

Just in case it remains, take a look at this bug if it's related. (incl. a screenshot of empty update-manager)
https://bugs.launchpad.net/ubuntu/+source/update-manager/+bug/588710