According to the documentation, it is possible to deploy to Azure by updating a git repository.
I have attempted the walkthrough here.
I created this github repository, then generated an ASP.NET MVC project from the Visual Studio template.
Looking at the logs, Azure detected the checkin, but provided this unhelpful message:
Using the following command to generate deployment script: 'azure site deploymentscript -y --no-dot-deployment -r "D:\home\site\repository" -o "D:\home\site\deployments\tools" --basic'.
Generating deployment script for Web Site
Generated deployment script files
Found solution 'D:\home\site\repository\kudu-deployment-test.sln' with no deployable projects. Deploying files instead.
Why is my straight-out-of-the-box ASP.NET project not a 'deployable project'?
What can I do to fix it so it is?
Your solution file as it currently exists in your GitHub repository does not contain any project information as the error suggests. I downloaded a copy of your repo and opened the solution in Visual Studio 2015 and it did not have any projects associated with it.
https://github.com/AndrewShepherd/kudu-deployment-test/blob/master/kudu-deployment-test.sln
Once I added the existing project from the files in the repository, this is what my updated solution file looked like:
If your project is correctly visible in Visual Studio, I would make sure that your solution file is saved and the updated version is pushed to your repo. Since you have already setup continuous deployment from GitHub, once you fix the solution file, your project should automatically deploy correctly.