I may have found an answer:
http://stackoverflow.com/questions/3475 ... g/34756693
I do not have time to test this right now, but it is interesting that these questions are being asked by others. If this is true, then we must forget what we already know about Windows networking APIs and use completely different network APIs in Windows Universal apps, including Windows IoT on the Raspberry Pi.
Replying to my older post, the above seems to be a dead end. The closest one gets to a ping functionality is trying to open a TCP connection to a port on the host. Of course, this works only if a process on the host is listening to the same port (e.g., 80 for HTTP) and accepts the connection. This is not what I need, but in the lack of anything else it might suffice if the goal is occasionally verifying that Internet connectivity exists. However, it would be wasteful to download the Microsoft, Google or Youtube home page every 5 minutes just to check for Internet connectivity. One's ISP's gateway would also be a primary target for pinging for this purpose, but these gateways rarely run publicly accessible HTTP servers or similarly accessible TCP/UDP services.
The general consensus I found is that ICMP is not available from a Windows Universal app. However, there are "Ping" apps in app store that promise some sort of connectivity test. I don't know if they provide true ICMP or just connection to a TCP or UDP socket on the host, with the limitations mentioned above.