*Hits self over head with Linux manual* Looks like I should have tested my original answer better before posting. If you do as I suggested, it will open new mozilla windows when mozilla is running but will no longer be able to start mozilla.
Below is a script (yes, this time I tested it, but still no guarantees) that I have placed in /usr/local/bin/mozilla that works for me. You may need to edit it to tell it where mozilla is on your system.
# Run Mozilla, but don't put up profile manager for new windows.
MOZILLA_OPTIONS=$* #Whatever was passed to the script will be passed to Mozilla
if ps -e | grep mozilla-bin > /dev/null
echo "Mozilla is already running, opening a new window."
MOZILLA_OPTIONS="$MOZILLA_OPTIONS -remote \"xfeDoCommand \(openBrowser\)\""
echo "Mozilla is not running, starting it now."