I used to have a script to shut down whatever service there was for Syndicate, because the site was constantly locking me out of the game for not playing often enough. I no longer have that issue, but the script was nothing special. Just had "Process, Close XXX.exe", but can also be "Run "[path to batch file] for an option.
The launcher should completely shut down when you exit the game. If not, confirm that Windows is not set to "always run" on start-up and check the settings within the games that use Rockstar. I only have Max Payne 3 & L.A. Noire games that use the launcher and I have a script to disable internet connection, as I got tired of messing around with "must update to run Max Payne 3" and then would still give me trouble.
L.A. Noire has not been an issue, but neither of these games continue to run Rockstar when exited.
Check your settings!
EDIT: On MP3, the updates were ONLY for Rockstar were all the stupid ass Social club crap, as the game plays 100% fine when the connection is gone. I don't play on-line games, so have ZERO use for any game that requires mandatory connection to the internet. Fortunately, only a few of my games require internet and have worked out all the bugs.