It depends on the complexity of the PCI Ethernet card, the availability of reference material (datasheets, open source drivers, etc) of the ICs used on the card, and your ability to adjust for that.
So possible, yes. For practical purposes, no, it's just not practical. But it can be done. For example, here is a project where someone uses an ISA bus (which PCI replaced) Ethernet Card, with a MicroChip Pic microcontroller. (some more info).
As far as I know, there is no Arduino to PCI bridge, or a project that specifically uses a PCI ethernet card, though some ethernet cards use the same chips that are used for Arduino Ethernet Shields, like the Microchip enc28j60.