Azure does not allow me to ping my hosted websites so what can I do with NetPath?
Netpath does not use ICMP for the end target, rather makes a TCP port connection to the address you supply. Therefore use the port of the service you are presenting out of Azure to allow NetPath to work to that target.
It does use ICMP as part of the probing methods for hops along the path however, but these show up into Azure dependant on your security definition. Even if they don't allow ping to devices along the path NetPath will still work.
