How to implement serial-to-fiber optic Ethernet with WIZnet W5500?
Since that device includes a PHY for wired ethernet over twisted pair copper, chances are you can't directly attach it to anything but twisted pair copper ethernet. The datasheet lacks any reference to standards, so it's very likely it's been tuned to exactly and exclusively that purpose. Best thing you can do is build it for wired ethernet, than use one of the many "media converter" boxes that translate between twisted pair ethernet and one of the fiberoptical ethernet standards.

So, you mention that you want to build a network-to-serial adapter with a specific networking interface.

That's a pretty normal problem, and that's why there's standard interfaces for connecting processors / microcontrollers to dedicated PHY chips. The standard you're looking for is "Media Independent Interface" (MII) or the "Reduced Media Independent Interface" (RMII). There's many microcontrollers that have that, and whose manufacturers have example implementations of Ethernet and TCP/IP stacks (gut feeling: you don't even need IP, and could probably work on Ethernet frame level just as well). Buy one of those, take an example firmware, connect a (R)MII PHY to the microcontroller's (R)MII pins, and be done. You can probably buy some eval board that does just that. 

Another alternative would of course be, assuming your serial port is slow (compared to the Gigabits you can push through optical cable) to not have the network logic on the serial side, but on the other end of fiber – so it'd be

Serial device – "dumb" signal translator – optical cable – "dumb" signal translator – serial-to-Network