Magento – Magento 2 : how to change order status processing to complete using Rest Api

Take a look at http://www.mexbs.com/magento-order-statestatus-flow/

Order "Pending" From the Admin panel, the sales order appears in the Order list as "Pending". At this point, payment has not been processed, and the order can still be canceled.

Payment Received The status of the order remains "Pending" until payment is received. Depending on the payment method, you, may be notified when the transaction is authorized and in some cases, processed. The status of the order is now "Processing" and order can not be cancel and you will need to issue a credit using Credit Memos see http://www.magentocommerce.com/knowledge-base/entry/order-process-overview

You can use the salesOrderManagementV1 to add a comment to an order email and send an email:

POST /V1/orders/{id}/comments
POST /V1/orders/{id}/emails

Some details from the code:

<route url="/V1/orders/:id/emails" method="POST">
    <service class="Magento\Sales\Api\OrderManagementInterface" method="notify"/>
    <resources>
        <resource ref="Magento_Sales::sales" />
    </resources>
</route>

You can also add a comment in that email:

<route url="/V1/orders/:id/comments" method="POST">
    <service class="Magento\Sales\Api\OrderManagementInterface" method="addComment"/>
    <resources>
        <resource ref="Magento_Sales::sales" />
    </resources>
</route>

Digging into the API code:

/**
 * Adds a comment to a specified order.
 *
 * @param int $id The order ID.
 * @param \Magento\Sales\Api\Data\OrderStatusHistoryInterface $statusHistory Status history comment.
 * @return bool
 */
public function addComment($id, \Magento\Sales\Api\Data\OrderStatusHistoryInterface $statusHistory);

/**
 * Emails a user a specified order.
 *
 * @param int $id The order ID.
 * @return bool
 */
public function notify($id);