Is there a way to install either the Android Market or Google Play on a Kindle Fire (Android 2.3 and Kindle version 6.3.1) without first rooting it? I have access to a Windows 7 computer with Admin privileges, but no USB cable (I do have ES File Explorer running an FTP server, so I can easily send and receive files via FileZilla on the computer).
You don't need to root your Kindle Fire for this -- as long as the following source is still valid: Install Android Marketplace and Google Apps on Amazon Kindle Fire. As that howto speaks about "Android Marketplace" (and not "Google Play"), it may not be up-to-date -- but I doubt the change of the name affected the procedure of installation.
A more up-to-date tutorial is Install Android Play Store on Amazon Kindle Fire - The Easiest Way, but that involves rooting the device. Maybe it has to do with the framework and a newer Android version, which makes rooting a necessary step.
EDIT: You may also want to take a look at these video tutorials: How To Install Google Music / Play On Your Kindle Fire! and GApps on Kindle Fire Without Rooting. Looks like you can still have Google Play installed this way without rooting.
The necessary .apk files for Google Play Store, Services Framework, etc., must be installed in
/system/app and doing so requires root access. So, you must root the Kindle Fire in order to do so.
However, you don't have to leave the Kindle Fire rooted. After you've installed it, you can always unroot the device (though I'm not sure why you'd want to do that).