DAVdroid does not determine whether there is an Internet connection. That’s all up to Android’s sync logic, which calls DAVdroid.
Did you have a look at FAQ: Why is synchronisation not triggered as set it in sync interval settings? Did you disable app standby for DAVdroid? Which Android version do you have? Which device do you have? Is there any additional information which could be useful for debugging?
Also, please provide steps to reproduce and debug info.