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:
In order to identify public IPv6:
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 126.96.36.199
This might be helpful when using console on some *NIX systems or on the routers.