1) use g:datePicker undocumented default="..." attribute in the view

<g:datePicker name="myDate" value="${new Date()}" noSelection="['':'-Choose-']" default="none"/>

2) remove the field from the map used for binding if some/all fields is/are empty

def excludes = []
if (!params.myDate_day)
excludes << "myDate"
bindData(myDomainObject, params, excludes)

(note that this has to be done in both save and update and that bindData has to be used rather than properties assignment)

I have:

<g:datePicker precision="day" name="publicLiabilityExpiry" value="${contractor?.publicLiabilityExpiry}" default="none" noSelection="${['':'--']}"/>

and need to use Option 2 to get it to work.