Azure web role supports https with node.js, but do Azure websites support https?
As of this post : link and as far as I'm aware it's still not possible to set up a custom SSL certificate at this time for an Azure Website.
If this is part of your requirement I would suggest going for a regular WebRole and setting up a Certificate to be used in the RoleConfiguration.
You can associate a certificate with a WebRole under the EndPoint configuration when the Protocol is set to https shown below:
Yes, this should work by default for non-custom domains (foo.azurewebsites.net). As @f0x mentioned, SSL support for custom domains isn't available for Windows Azure Web Sites.