https://learn.microsoft.com/en-us/azure/nat-gateway/troubleshoot-nat
as you can see NAT gateway supports IPv4 UDP and TCP protocols. ICMP isn't supported and is expected to fail.
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I am unable to ping any public website like google/yahoo from my private azure VM where traffic is routed through NAT Gateway.
I am able to browse the website from browser but unable to ping.
If NAT gateway is not supporting ICMP protocol then please share the official documentation for the same.
https://learn.microsoft.com/en-us/azure/nat-gateway/troubleshoot-nat
as you can see NAT gateway supports IPv4 UDP and TCP protocols. ICMP isn't supported and is expected to fail.