Well if it's a server, things like "256 nVidia Geforce 7300LE TurboCache" are really not necessary (servers don't even need screens, they can be administered using other means if necessary; the less hardware, the less possible things that might break down). But to your question.
Fedora Core probably makes a good server, but it has quite a lot of software, i.e. a lot of possible security holes. I myself would prefer taking a distribution that has a minimum amount of software installed, just the ones that you really need; that way keeping it up to date, securing it and using it is easier (resources). Slackware is probably a great choice, if you can configure it to your needs. During setup, don't install anything you don't need.
I don't think it's really up to which distribution you pick up, any one of them can probably be made a good server. It's just the amount of configuration that you need to do before it's up and running -- security matters, hardening, possible kernel and/or drivers compilation for hardware that is not supported by the stock kernel (well, the kernel shouldn't have any unneeded stuff in it either) etc.. Fedora very probably works with your hardware out-of-the-box, or at least better than some extremely small distribution or one that is aimed at certain purpose only, but at the cost that you get more than you ask for, and waste harddisk space. Actually for a working server you don't need much more than security updates and what they need; "don't fix a working thing". This is why you shouldn't even update software that is known to work, unless you have a good reason to do so (like security updates and fixes). Having extra software (that is not needed) on board means having software that has possible security holes, and having files that you might backup even if you didn't need to, and..well, you just don't have a reason to have software that the server does not need (like KBounce).
Consider the stability of the system too; is Fedora Core or Slackware generally considered more stable than the other one? Why? Could you fix it to get a more stable system?
Fedora Core (I've been testing it now for a while, but only on desktop use) eats up somewhat much resources, but on the other hand I haven't tuned it much (I have no reason for now). But still it provides server stuff which is pretty easily available, and if you have time to trim it, take the unneeded software off it, harden it and configure it well, it's probably a good choice.
fedora is *NOT* a server distribution. try centos instead if you're used to fedora.
I disagree. It is not
up to the distribution, since nobody in his/her full understanding will put it do server job straight out of the box. And if configured, any distribution can be made whatever needed; the kernel is Linux anyway, if you want Linux, and the rest is software, init scripts, file system choices etc.. And since Fedora is a "test field" for those RH Enterprise Linuxes, it surely works as a server -- given enough work is made to turn it into a server.
Windows Server 2003 is the same thing; I wouldn't pick it up to be a server, but it can be used as one (even pretty well).