Use part of Date field for grouping / sorting in View
Go to your view and add an extra date field, but hide it from display. Use that as the grouping field. Then create a views template for that field. You can look at the template name suggestions, something like views-view-field--field-date-1.tpl.php would work

The default content of that file would be 

&lt;?php print $output; ?&gt;


But you need to replace that with

&lt;?php print date('F Y', strtotime($row-&gt;node_data_field_date_field_date_value)); ?&gt;


Where 'F Y' is the PHP date string format http://php.net/manual/en/function.date.php