I faced the same problem and the solution is to add the user that starts the ArcGIS Server service to the SQL Database. Say that .\arcgis is the ArcGIS Service user that is shown in the services window (services.msc command). You have to add this same user to the SQLGeodatabase.
To do this you only need to execute the "Add User" tool. This tool either in the toolbox, or right click in the geodatabase\Administration\Add user. There you have to put NameOfYourServer\arcgis as the user to create. In my case was like this: