I have a Dell 18" at work, with a 128 MB GeForce Ti4200 using the DVI connector. Dell has their monitors manufactured by other people, so the different models are not necessarily comparable. Mine is made by LG (the former Lucky-Goldstar in S. Korea). It has 1280x1024 pixels. The 19" has the same number of pixels, as far as I know. The Dell 20" has a 1600x1200 screen.
I've run GalCiv v1.1 full screen on the 18" (just for a few minutes - I haven't played it at work in a while). If the resolution of the input signal is not 1280x1024, the monitor is responsible for scaling the image, which introduces fuzziness; other than that, it has always worked fine for me.
So, GalCiv does work with LCD monitors of your size. Furthermore, I'd say any issues are related to your video card and/or monitor since they are responsible for showing images of any size.
LCD monitors require a 60Hz refresh rate. Since the game is probably running at a different resolution from your desktop, you need to make sure that that resolution is set to 60Hz on your machine. This can be a real pain - you may need to download some third-party utility to get this done. Also, make sure you have installed the monitor "driver", an INF file that describes the capabilities of the monitor. Windows Update may supply such a thing.