I can only comment this from the SQL Server side of things. The error message means that you are not making contact. This could be because the instance is not running, does not accept remote connections or there is a firewall in the way. It could also be that the server or instance name is wrong, or in case of a named instance that the browser service is not running or inaccessible due to a firewall.
It appears that you seem uncertain how to configure the connection settings for AppService (with which I am completely unfamiliar with). To make the equation easier, try to connect to SQL Server from something else on the same machine, preferably a tool like SQLCMD. If you can get that connection to work, the problem has to be in the AppService configuration.