Use the global constant DATETIME_DATETIME_STORAGE_FORMAT to format the date:
// If the time is in a different timezone then adjust it to UTC before you save it
$date->setTimezone(new \DateTimezone(DATETIME_STORAGE_TIMEZONE));
$paragraph = Paragraph::create([
...
'field_date' => [
'value' => $date->format(DATETIME_DATETIME_STORAGE_FORMAT),
],
]);
Edit
For Drupal >= 8.5: Global constants in datetime.module are deprecated and DateTimeItemInterface has been introduced.