It looks like Valve has open-sourced their peer to peer networking code, and it is capable of running without Steam. I think it accepts an IP address and port number to connect to:
https://github.com/ValveSoftware/GameNetworkingSockets

Can you recommend a library for managing a public game server list? I could write a basic one myself but if there is a more robust lib out there that handles this I would prefer to use it.

Just a list of servers isn't particularly helpful.

You typically want the list to keep refreshing with available servers (so stopped servers fall off,) as well as use this list to do NAT punch-through. Very often, you will also want to do “game type filtering” for people looking for “deathmatch” versus “co-op” or whatever.

You might want to look at available STUN servers, because they solve the NAT part of the equation.