You can do that with a custom SQL query, something along the lines of:
global $wpdb;
$results = $wpdb->get_results( "SELECT DATE(post_date) AS d, COUNT(*) AS c FROM $wpdb->posts WHERE post_status="publish" GROUP BY d" );
You’ll get an array of objects where d is the date and c is the number of posts published on that date. You should also put LIMIT constraints if you plan to query large datasets.
Hope this helps.