Okay, I can confirm this would appear to fix the problem.

Test procedure:

1. Log in remotely to the affected machine using ssh, run the following command:

   $ sudo watch ls -l /proc/$( pidof gdm-binary )/fd

2. Start a remote X session with that host (through Xnest, plain X, whatever)... observe that new files are opened.

3. Close the X session, observe that the files are not removed from the list.  In my case, this looked like this:

Vanilla Ubuntu gdm build.
After initial start-up.

total 0
lrwx------ 1 root root 64 2012-09-20 14:34 0 -> /dev/null
lrwx------ 1 root root 64 2012-09-20 14:34 1 -> /dev/null
lrwx------ 1 root root 64 2012-09-20 14:34 2 -> /dev/null
lrwx------ 1 root root 64 2012-09-20 14:34 3 -> socket:[78942]
lr-x------ 1 root root 64 2012-09-20 14:34 4 -> pipe:[78945]
l-wx------ 1 root root 64 2012-09-20 14:34 5 -> pipe:[78945]
lr-x------ 1 root root 64 2012-09-20 14:34 6 -> inotify
lr-x------ 1 root root 64 2012-09-20 14:34 7 -> pipe:[78952]
l-wx------ 1 root root 64 2012-09-20 14:34 8 -> pipe:[78952]
lrwx------ 1 root root 64 2012-09-20 14:34 9 -> /var/run/gdm/auth-for-gdm-omI8J6/database
lr-x------ 1 root root 64 2012-09-20 14:34 10 -> pipe:[78971]
l-wx------ 1 root root 64 2012-09-20 14:34 11 -> pipe:[78971]
lr-x------ 1 root root 64 2012-09-20 14:34 12 -> pipe:[78937]
lrwx------ 1 root root 64 2012-09-20 14:34 13 -> socket:[78973]
lrwx------ 1 root root 64 2012-09-20 14:34 14 -> socket:[78974]

After a few logins...

total 0
lrwx------ 1 root root 64 2012-09-20 14:38 0 -> /dev/null
lrwx------ 1 root root 64 2012-09-20 14:38 1 -> /dev/null
lrwx------ 1 root root 64 2012-09-20 14:38 2 -> /dev/null
lrwx------ 1 root root 64 2012-09-20 14:38 3 -> socket:[78942]
lr-x------ 1 root root 64 2012-09-20 14:38 4 -> pipe:[78945]
l-wx------ 1 root root 64 2012-09-20 14:38 5 -> pipe:[78945]
lr-x------ 1 root root 64 2012-09-20 14:38 6 -> inotify
lr-x------ 1 root root 64 2012-09-20 14:38 7 -> pipe:[78952]
l-wx------ 1 root root 64 2012-09-20 14:38 8 -> pipe:[78952]
lrwx------ 1 root root 64 2012-09-20 14:38 9 -> /var/run/gdm/auth-for-gdm-omI8J6/database
lr-x------ 1 root root 64 2012-09-20 14:38 10 -> pipe:[78971]
l-wx------ 1 root root 64 2012-09-20 14:38 11 -> pipe:[78971]
lr-x------ 1 root root 64 2012-09-20 14:38 12 -> pipe:[78937]
lrwx------ 1 root root 64 2012-09-20 14:38 13 -> socket:[78973]
lrwx------ 1 root root 64 2012-09-20 14:38 14 -> socket:[78974]
lrwx------ 1 root root 64 2012-09-20 14:38 15 -> /var/run/gdm/auth-for-gdm-bqKAEb/database
lrwx------ 1 root root 64 2012-09-20 14:38 16 -> /var/run/gdm/auth-for-gdm-QC37Jk/database
lrwx------ 1 root root 64 2012-09-20 14:38 17 -> /var/run/gdm/auth-for-vrtadmin-4oPw6L/database
lrwx------ 1 root root 64 2012-09-20 14:38 18 -> /var/run/gdm/auth-for-gdm-ePg83T/database
lrwx------ 1 root root 64 2012-09-20 14:38 19 -> /var/run/gdm/auth-for-vrtadmin-YMcsem/database

Now, build gdm with the given patch here, re-start gdm, and try again with the same procedure.  In my case, I see:

Patched gdm build.
After initial start-up.

total 0
lrwx------ 1 root root 64 2012-09-20 14:49 0 -> /dev/null
lrwx------ 1 root root 64 2012-09-20 14:49 1 -> /dev/null
lrwx------ 1 root root 64 2012-09-20 14:49 2 -> /dev/null
lrwx------ 1 root root 64 2012-09-20 14:49 3 -> socket:[102450]
lr-x------ 1 root root 64 2012-09-20 14:49 4 -> pipe:[102453]
l-wx------ 1 root root 64 2012-09-20 14:49 5 -> pipe:[102453]
lr-x------ 1 root root 64 2012-09-20 14:49 6 -> inotify
lr-x------ 1 root root 64 2012-09-20 14:49 7 -> pipe:[102460]
l-wx------ 1 root root 64 2012-09-20 14:49 8 -> pipe:[102460]
lrwx------ 1 root root 64 2012-09-20 14:49 9 -> /var/run/gdm/auth-for-gdm-kjZfwe/database
lr-x------ 1 root root 64 2012-09-20 14:49 10 -> pipe:[102479]
l-wx------ 1 root root 64 2012-09-20 14:49 11 -> pipe:[102479]
lr-x------ 1 root root 64 2012-09-20 14:49 12 -> pipe:[102445]
lrwx------ 1 root root 64 2012-09-20 14:49 13 -> socket:[102481]
lrwx------ 1 root root 64 2012-09-20 14:49 14 -> socket:[102482]

During a remote X session
total 0
lrwx------ 1 root root 64 2012-09-20 14:51 0 -> /dev/null
lrwx------ 1 root root 64 2012-09-20 14:51 1 -> /dev/null
lrwx------ 1 root root 64 2012-09-20 14:51 2 -> /dev/null
lrwx------ 1 root root 64 2012-09-20 14:51 3 -> socket:[102450]
lr-x------ 1 root root 64 2012-09-20 14:51 4 -> pipe:[102453]
l-wx------ 1 root root 64 2012-09-20 14:51 5 -> pipe:[102453]
lr-x------ 1 root root 64 2012-09-20 14:51 6 -> inotify
lr-x------ 1 root root 64 2012-09-20 14:51 7 -> pipe:[102460]
l-wx------ 1 root root 64 2012-09-20 14:51 8 -> pipe:[102460]
lrwx------ 1 root root 64 2012-09-20 14:51 9 -> /var/run/gdm/auth-for-gdm-kjZfwe/database
lr-x------ 1 root root 64 2012-09-20 14:51 10 -> pipe:[102479]
l-wx------ 1 root root 64 2012-09-20 14:51 11 -> pipe:[102479]
lr-x------ 1 root root 64 2012-09-20 14:51 12 -> pipe:[102445]
lrwx------ 1 root root 64 2012-09-20 14:51 13 -> socket:[102481]
lrwx------ 1 root root 64 2012-09-20 14:51 14 -> socket:[102482]
lrwx------ 1 root root 64 2012-09-20 14:51 15 -> /var/run/gdm/auth-for-gdm-CM7Tt4/database

After the remote session terminates.
total 0
lrwx------ 1 root root 64 2012-09-20 14:51 0 -> /dev/null
lrwx------ 1 root root 64 2012-09-20 14:51 1 -> /dev/null
lrwx------ 1 root root 64 2012-09-20 14:51 2 -> /dev/null
lrwx------ 1 root root 64 2012-09-20 14:51 3 -> socket:[102450]
lr-x------ 1 root root 64 2012-09-20 14:51 4 -> pipe:[102453]
l-wx------ 1 root root 64 2012-09-20 14:51 5 -> pipe:[102453]
lr-x------ 1 root root 64 2012-09-20 14:51 6 -> inotify
lr-x------ 1 root root 64 2012-09-20 14:51 7 -> pipe:[102460]
l-wx------ 1 root root 64 2012-09-20 14:51 8 -> pipe:[102460]
lrwx------ 1 root root 64 2012-09-20 14:51 9 -> /var/run/gdm/auth-for-gdm-kjZfwe/database
lr-x------ 1 root root 64 2012-09-20 14:51 10 -> pipe:[102479]
l-wx------ 1 root root 64 2012-09-20 14:51 11 -> pipe:[102479]
lr-x------ 1 root root 64 2012-09-20 14:51 12 -> pipe:[102445]
lrwx------ 1 root root 64 2012-09-20 14:51 13 -> socket:[102481]
lrwx------ 1 root root 64 2012-09-20 14:51 14 -> socket:[102482]

For those who get this problem and want a package; download the patch, then:

$ sudo apt-get build-dep gdm
$ apt-get source gdm
$ cd gdm-2.30.2.is.20.30.0
$ cp ../gdm-2.30.2.is.2.30.0-filehandle-fix.diff debian/patches/
$ dpkg-buildpackage