can you download/install other apps? did you back out & retry? powercycled the tablet? have you uploaded or downloaded new files to the tablet? I found a video file I uploaded from my PC to my Tab was causing applications not to install on the Market. When removed file was able to download and install apps again.
I just installed flash. It's weird the way it installs it. I also thought mine wasn't installing. I downloaded it, and when I clicked on OPEN button, a screen popped up telling me full access is allowed. I clicked OK, and the flash app was put in the uninstalled section. When I clicked the install button, the same screen would pop up telling my full access is allowed and only gave me the OK button. I thought it was weird. I rebooted and the flash app was no longer in the uninstalled section. I checked my alerts and I had 2 alerts. 1 saying I had an update available for a game and the other alert said flash was installed successfully. I was able to watch youtube videos no problem.
Are you getting any error messages or nothing happens when you try to install it?
I had no problems with youtube videos and the native app. Restarted and flash did as you described, but it seems firefox doesn't recognize that it is installed, but the stock browser does. Will try reinstalling Firefox next.