I'm new to Civ 3, so please be patient with me. I've been building the spaceship and everytime a new component is built I get a screen showing me my progress and an option to launch. So far so good. I build the last component and I don't get that screen and I can't figure out how to launch the darn thing. Is this a bug or am I missing something?

It should launch automatically once you finish the final piece. A window should come up and give you the button to launch. When that happens you have no choice but to launch your spaceship.
I think either you aren't done yet with your ship or there's an error in your game. If the window never popped up that points to an error, but I don't know which is the case.