You will not see the public ip in IIS binding because you don't have a NIC with an external IP connected to it. The cloud service is the one responsible to bridge your VM (or set of VMs) with the external world and it is also this service which will give you a public IP.
To get your public IP (VIP) just navigate to Cloud Services -> Check ip in quick glance (right side of the dashboard) >> PUBLIC VIRTUAL IP (VIP) ADDRESS
If needed, you can also have more than one VIP:
https://azure.microsoft.com/en-us/documentation/articles/load-balancer-multivip/
To access your website from the internet, you have to setup the VM endpoints:
https://azure.microsoft.com/en-us/documentation/articles/virtual-machines-set-up-endpoints/
Make sure Windows Firewall is not blocking the connections as well.
Update:
Reserved IP Overview
https://azure.microsoft.com/pt-br/documentation/articles/virtual-networks-reserved-public-ip/
Reserving a VIP
New-AzureReservedIP -ReservedIPName MyReservedIP -Location "East US"
Replace MyReservedIP for your desired name. Replace Location with your data center location. To get a list of locations, use Get-AzureLocation
Associate with your running Cloud Service
Set-AzureReservedIPAssociation -ReservedIPName MyReservedIP -ServiceName TestService
Replace MyReservedIP with the one you created and service name with your cloud service name.
How to install and configure Azure PowerShell
https://azure.microsoft.com/en-us/documentation/articles/powershell-install-configure/