Roman's answer worked fine for me - or at least, it gave me what I was expecting. Opening the video in the phone's native application is exactly the same as what the iPhone does.
It's probably worth adjusting your viewpoint and expect video to be played fullscreen in its own application, and coding for that. It's frustrating that clicking the video isn't sufficient to get it playing in the same way as the iPhone does, but seeing as it only takes an onclick attribute to launch it, it's not the end of the world.
My advice, FWIW, is to use a poster image, and make it obvious that it will play the video. I'm working on a project at the moment that does precisely that, and the clients are happy with it - and also that they're getting the Android version of a web app for free, of course, because the contract was only for an iPhone web app.
Just for illustration, a working Android video tag is below. Nice and simple.
<video src="video/placeholder.m4v" poster="video/placeholder.jpg" onclick="this.play();"/>