Yes, this is indeed possible with actions and templates.
The trick is to create a template where you use the \date
action in the title like this:
If you like you can further tweak the exact format of the date as described here:
Also, note that you can give the template a custom name by double-clicking the entry in the template list on the left:
This will then be the name you see in the list when holding the plus button, or in your case, when clicking a day header in the inspector:
Hope that helps!