How to enable Ping(ICMP) on Azure

I don't believe you can do this. Traffic leaving the data center goes through the load balancer, and the load balancer only routes TCP-based traffic.

I know this question is very old, but I stumbled upon it while facing the same issue and there is an actual solution for it now in Azure.

When setting up your Virtual machine you can assign it an "Instance IP address". Once that has been configured, you can enabled ICMP in and out in the local firewall. You will then be able to ping out of your Azure VM and also use tools like traceroute.