To monitor an Azure SQL Server using SQL Diagnostic Manager, you have to create a SQL Server user on the instance with the following minimum permissions:

Minimum permissions Azure SQL

• db_datareader - Grants the user the ability to read all data from all user tables.
  
• View Definition - Grants the user to see the metadata of the securable where the permission was granted.
  
• View Database State - Grants the ability to view the conditions of the database through the database-level dynamic management views or functions.
  
• Select - Grants the ability to use the SELECT command against all applicable objects within the database.
  
• Execute - Grants the ability to use the EXECUTE command against all applicable objects within the database.
  
• Connect - Grants the ability to enter the database. When a new user is created, this permission is granted by default.
  

Recommended Permissions Azure SQL

• Minimum permissions plus Control and db_datawriter.
  • Control - Grants permissions on the securable.
    
  • db_datawriter - Grants access to tables and views within a database.