Replace the hinge with another euro-style hinge. I don't know that specific one, but on some euro hinges you can see the spring that closes the door. In your photo I don't see that hinge, so maybe it broke or simply wore out and get stretched so it doesn't close the door any more.

If you measure the screw spacing and the cup size you can probably just replace the hinge for a few bucks. A new hinge should have a more positive snap closed action to it.