Is there a way to create an execute stored procedures on SQL Azure database using REST API or Azure Management Libraries?
I checked Operations for Azure SQL Databases here http://msdn.microsoft.com/en-us/library/azure/dn505719.aspx and Management Libraries for .NET here http://msdn.microsoft.com/en-us/library/azure/dn602775(v=azure.11).aspx, and I don't see any options.
No, this is something you would need to do in your application. The REST API provides management capabilities at the infrastructure/platform level. Procedure execution would fall under application level IMHO. Management Library for .Net is simply a wrapper around these REST API.
I think your only option would be write your own API specific for your application/database that can execute the stored procedure in your database.