There is no online game for GalCiv. Maybe it comes with an add on for GalCiv II.
But you can register your player here Link
and join the battle in the Metaverse Link
Basically it is a ranking list. Optional you can join an empire. The scores of the empire overlords are added and the empires fight for the domination of the galaxy.
When you finished a game on a computer which is online you are asked whether you would like to submit your score to the Metaverse. When you keep the Endgame.sav you can do it from another comuter at another time if you like.
e.g. you can join the Jedi Republic here Link after you registered your player. Currently we are number three in the empire ranking
Update: I just realized that you already founded your own empire. Good luck for you.