UPDATE: No you CANNOT watch prime videos on your Samsung Galaxy Tablet. Amazon doesn't publicize this, of course. But it is not an issue of Flash anymore. It is an issue of Amazon choosing not to support Samsung tablets. Samsung TVs work fine. I did have it working on my tablet. About 2 weeks later, Amazon pulled the plug. I don't know how they block it, but they do.
Need confirmation? Go to Amazon's own list of supported devices. Samsung tablets are not on the list of compatible devices.
I just got my Galaxy Tab 2 7.0 to stream from Amazon!
You have to go find a copy of Flash Player 11.1_11 and download it onto your tablet. I'm not going to give a URL, because I don't want to get any website in trouble. But the Flash Player is out there, just search the internet for it. You can't get it through Google Play.
Once you have downloaded it you need to go into the files on your tablet (I use OI File Manager for this) and double-click the flash file to install it. Your tablet may tell you that security settings won't let you install it. No problem. If you get this message, go into your Settings and click on Security. Under Device Administration click the box to allow installation of non-Market apps. You are taking a risk here. So it is your choice - but otherwise you can't stream video.
After you have fixed your security and gone back and installed your Flash Player download, open up your browser. Amazon is actually a good page to start with because they often have video clips on the front page. Firefox will show you an empty window where the video should be and prompt you to click (tap) to use the plugin. After that, I could even use the Tablet's version of Chrome to stream Amazon Prime.
That said, it seems to be Flash that is the problem - not which browser you have. I suspect that this is not Amazon's fault. It's bad business to not allow people to access your service on certain kinds of tablets. Maybe Adobe got mad and took it off the market.
The answer is YES, you need Flash installed, see here: http://forum.xda-developers.com/showthread.php?t=1774336 . Works with Firefox Beta and Flash 184.108.40.206, haven't tried Opera. Chrome doesn't support Flash. Install the Auto Full Screen Firefox add-on and you can watch videos in full screen without the nav bar.