UPDATE: It works now on my Galaxy Tab 4 8". Here's what I did:
-Uninstall the Amazon app I got from the Google Play app store.
-Install the Amazon app store by downloading the apk from Amazon.com using a web browser.
-Using the Amazon app store, install the Amazon app listed there and the Amazon Instant Video app.
Now, from the Amazon app, I can peruse the Prime Video selections, and watch free items.
A word of caution: after doing this, it enabled 1-click purchasing (it was previously disabled for my Amazon account), which I disabled, but forced on 1-click purchasing for digital items with no way to disable. Be careful what you click.