Eclipse Markdown Plugin - How to open Markdown HTML Preview when double clicking a .md file?

Perhaps this is not exactly what you've asked for, but if you open the Markdown HTML Preview view and leave it open, it will automatically change to whatever .MD file you are actually editing. So, just leave that View open and it will perform as you want. You can then save your Perspective to keep that preference (Eclipse does that for you anyway).

More, if you're working with GitHub Flavoured Markdown you'll want to check the GitHub Flavored Markdown viewer plugin, which actually works with the Markdown plugin, but does the rendering as you can see in GitHub. In this case, you'll actually have what you're looking for: if you right click any .MD file under your project explorer, there is an option to Show in GFM View, which will open that view in case it is not open already.

Update (September 2020)

There is one fork in Github that claims to be compatible with Eclipse 2020-03. I didn't test it myself, since I am not longer using Eclipse. Thanks to Daniel Armbrust who provided this link in the marketplace comments.

Open your README.md using the markdown editor (Generate Markdown Preview option) and open the markdown preview window: Window > Show View > Markdown View.

This will render your .md file.