No, Gradebook does not save automatically. Gradebook loads in a view-only state. If you want to make changes to scores or leave comments, you can do so by clicking an assessment and choosing "Edit Scores". However, once you've entered your scores and/or comments, you must save your work by clicking the green save icon. This needs to be done for each assessment you edit, as you edit them. This process helps to reduce the likelihood of unintentional changes made to student scores.
You can expect to see a warning message, similar to the one below, in the event that you're about to lose unsaved changes!