Only on azure: Could not create SSL/TLS secure channel

Capture the TLS handshake. If your ServerHello is missing you most probably don't have a common cipher suite with the remote.

Run both through https://www.ssllabs.com/ssltest/ to check supported cipher suites at both ends. For Windows Server, cipher suites can only be enabled or disabled globally (as in no distinction between client/server component) so that's why this makes for a good test.

UPDATE: Found a glaring problem in my reasoning, App Service has a frontend layer and that's where TLS terminates, so comparing ciphers that way goes nowhere.

Instead, run

Get-TlsCipherSuite

from Kudu's PowerShell and compare ciphers against your remote API's (the ciphers of which you can check over at https://ssllabs.com/ssltest). You should have at least one match.

If none match, you'll need to switch to Cloud Services or VMs and enable at least one of the cipher suites your remote speaks. Having to go this direction usually means one thing -- your remote is using weak cryptography (SSL 3.0 or TLS 1.0 with RC4) and you should have a chat with those citizens, or find new citizens that are rocking TLS 1.2.

From your System.Net trace:

[8356] 00000000 : 15 03 03 00 02

That's byte sequence for Fatal Handshake Error, which builds on my no common cipher theory.

Note the first byte (0x15):

 Record Type Values       dec      hex ------------------------------------- CHANGE_CIPHER_SPEC        20     0x14 ALERT                     21     0x15 HANDSHAKE                 22     0x16 APPLICATION_DATA          23     0x17

I ran across this error when hosting a client's certificate on Azure's App Services. The fix was to set WEBSITE_LOAD_CERTIFICATES in the App Settings.

You can set it as " * " to allow all certificates, or you can define specific certificate thumbprints to allow. See more info here.