if i correctly undersatnd your question..
you want count that how many call attented on peticular DATE
try below query:-
SELECT count(*) as '22nd june' FROM Process_Details
WHERE (SubSite_Id = @SubSite_Id) AND (Call_Attended = @callattend)
AND CONVERT(VARCHAR(10),call_start,101) = CONVERT(VARCHAR(10),@callstart1,101)