custom domain does not route to index.html
An assigned custom domain does not route to index.html similar to the original Azure-assigned URL.
Houston Haynes commented
Correction: index.html is automatically applied to designated "core.windows.net" domain. But the custom domain takes the value that's explicitly entered in the "index document name" field. I took it to mean that since I saw an italicized "index.html" that it was a default value. I got this impression due to the fact that the page resolved automatically with the default domain. Once I added an explicit "index.html" to the field then it *also* resolved for the custom domain.
Either change the prompt in the field to be an italicized "example: index.html" (and require the user to input a distinct value for that field) or make index.html the default for both the original and custom domains.