UPDATE as of 17-APR-2018 you can, read about that here.
Short answer is, prior to 17-APR-2018, that you couldn’t. (see alternative solution below) The reason is that when you deploy an Azure App Service it goes into a multi-tenant scale unit. A scale unit looks something like Figure 1, which I stole from here. Also, have a look at this forums discussion here.
Figure 1, Azure App Service scale unit, image taken from here
This scale unit has multiple customers running on it (i.e. multi-tenant). As SSL communication is offloaded on the Front Ends the only way to disable TLS 1.0 or TLS 1.1 (not supported) would be to make that configuration on the Front End, but then that would disable it for all other App Services running in that scale unit. Some customer may want it enabled for legacy reasons. NOTE: When you run in Basic, Standard or Premium mode the Web Worker in Figure 1 represents a virtual machine which is dedicated to the App Services running in your App Service Plan see here for more.
The solution is to get a scale unit all for yourself which is called an App Service Environment (ASE), see here. This gives you a scale unit and you can disable TLS 1.0, you cannot disable TLS 1.1 at the moment. The ASE also allows you to setup NSG and have more control over the feature.
You will need to access the resource explorer here and modifying the clusterSettings entry for the ASE, similar to the following and also discussed here.
UPDATE #1: If you happen to get something like ” ‘Code’: ‘BadRequest’, ‘Message’: ‘An error occurred when updating the entry :” when trying disable TLS 1.0 on and App Service Environment, in resource explorer, in the same context as you where in when you received the error, look for an item named “dnsSuffix: <ASEName>.p.azurewebsites.net” and delete it. Then retry.
UPDATE #2: Apparently you can also try to put an Application Gateway in front of the App Service, but what I know is this only possible in front of an ILB ASE, but if you are going to deploy an ASE you wouldn’t need the AG to disable TLS 1.0. See more about that here. Set-AzureRmApplicationGatewaySslPolicy -DisabledSslProtocols TLSv1_0, TLSv1_1 -ApplicationGateway $gw
UPDATE #3: I am not a PCI compliance officer and I am not clear on the date when the having TLS 1.0 enabled voids PCI compliance. However, it is my understanding that Azure App Services is PCI compliance and the removal or the ability to remove TLS 1.0 will happen before the compliance is voided. See here and here for some more information.