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

This might be helpful when using console on some *NIX systems or on the routers.