If you need to identify your public IP and have no possibility to run web browser of any kind you can use telnet.
In order to identify public IPv4:
telnet v4address.com
In order to identify public IPv6:
telnet v6address.com
Once connection is established autoresponder will show your public IP address:
This is the telnet autoresponder at v6address.com. You have connected over IPv4. Your IP address is 5.10.15.20
This might be helpful when using console on some *NIX systems or on the routers.