Cadaver747: Thank you, much appreciated!
StingingVelvet: Fair enough, I was just going by what people said above. It would be pretty odd for Origin to do that, but it also wouldn't really surprise me either. In any case, easy solution is easy (and also DRM free matters more for preservation than for avoiding using a client you already use).
So as I suspected, with Origin installed, active and logged in, Running Dead Space made no calls to Origin, or required origin to run. So then I disabled Origin, and the game runs fine. Then I uninstalled, and installed via Origin. The game plays fine. I disabled Origin, and started the game, and it does indeed call Origin.
So then I uninstalled the Origin version and installed the GOG version again, and started it with Origin closed. The game did not call Origin. I then booted Origin and tried to run it again, and Origin detected the game and asked if I accepted the EULA, and wanted to update. However, cancelling this worked fine and I was able to play the game without incident. I suspect the game isn't working on the OP's system for some reason, and he is running the game with Origin open. I think these two issues are unrelated but concurrent and it coincidentally seems as though the game will not run until the EULA is accepted.
Running the game with Origin closed after declining the EULA and update did not call Origin to reopen.
Just a PSA to anyone out there though, make sure if using Galaxy to have GOG selected for the DRM Free version of the game (see attachments)