Hi @Florian,
Have you checked this How to Create an "Add to Calendar" Button for Events - may be this will work for you?
As a side note - you can’t access users’ calendars directly unless you know their calendar address; the feature you’ve described is managed by mail clients (like Gmail which recognizes an .ics attachment and puts it to your calendar).
Also as I remember that @James_App_Maker was doing something with Calendar invites.
Hope this helps.
Best regards, Victor.