How can I create a list of page titles from custom meta values?

You need to get the title for each post ID in your array. The straightforward way is this:

$procedure_title_list = array();
$procedure_list = get_post_meta($patient_story->ID, 'Procedure', false);
foreach ( $procedure_list as $procedure ) {
    $procedure_title_list[] = get_the_title( $procedure );
}
echo implode( ', ', $procedure_title_list );