If you do pcimodules it should output all the modules available. You wouldn't download the driver because the driver should already be present in the kernel. I don't know whether or not fedora has the driver compiled in or as a module, but it depends on the person configured the kernel.
OH.. I'm sorry, I read your post wrong, I thought it was regular old PCI. r8169 would be the PCI version of your card, so heh... it wouldn't work.
Could also try modprobe r1000?
But I wouldn't know how to install it.