Azure Maps doesn't provide a service for this currently, however this may a part of the Azure Maps Creator platform in the future.
One option you could look at is putting the tiles in blob storage and then using it's authentication. If your map tile set is in the hundreds of thousands of tiles or less, this will likely work well. Be sure to enable CORs on your blob storage so that the tiles can be accessed in client web apps. I've used this approach for this sample tile set: https://azuremapscodesamples.azurewebsites.net/?sample=Tile%20Layer%20Options
If you have millions of tiles, then a tile server would likely be needed to maintain performance.