Oops.
There's no Google Play app on the Amazon Fire TV Stick.
I found this post explaining, Use this trick to watch Google Play movies on your Apple TV.
The tips work for the Amazon Fire TV Stick too.
1. Download the YouTube.com app.
2. Open the Amazon Fire TV Stick YouTube.com app.
3. Authorize via web.
4. Then go to the YouTube.com app menu and choose Library.

Image: YouTube.com menu - Library
5. Then scroll down to Purchases.

Image: YouTube.com menu - Purchases
There's your purchased Google Play movie!
Update:
It won't play!

Wait! What?
Is it just this movie?
Let me know your experience.