You should pass a raw email to \Google_Service_Gmail_Message() class.
This raw message should be a mime message base64 encoded. Pretty much everything you did.
BUT: you need not to pass the email body (like the content of the email address), but literally the whole email - with headers. So not $_POST["message"]
should be passed, but a prepared email that has all the required headers, including TO, FROM etc. Here is an example:
From: [email protected]
To: [email protected]
Subject: YOUR CUSTOM SUBJECT
Content-Type: text/plain; charset=UTF-8
Mime-Version: 1.0
Content-Type: multipart/alternative; boundary=5ae50a40c56153a0ca99d08aaf8f99d53f63981962d4f199b2339614636b
--5ae50a40c56153a0ca99d08aaf8f99d53f63981962d4f199b2339614636b
YOUR CUSTOM EMAIL CONTENT GOES HERE. Might be a mix of html and plain text.
--5ae50a40c56153a0ca99d08aaf8f99d53f63981962d4f199b2339614636b--
It's much better to use something like PHPMailer class to prepare the whole MIME version of the email for you, where you just pass to its public methods required data, subject, FROM, TO, body etc - and then request a MIME with a single method like this $phpmailer->getSentMIMEMessage()
.