Ho fatto cosi ma non mi da nessuna IP:
codice:
function TForm1.GetPublicIp: String;
    var
      IP: String;
      begin
      IdHttp1.Request.Host:= 'http://www.whatismyip.com';
      IP := IdHttp1.Get('http://www.whatismyip.com/automation/n09230945.asp');
end;

procedure TForm1.Button1Click(Sender: TObject);
begin
  ShowMessage(GetPublicIp);
end;