Python vs Groovy vs Ruby? (based on criteria listed in question)
This sort of adding-up-scores-by-features is not a good way to choose a programming language. You'd be better off choosing whichever you know the best. If you don't know any of them, try them out for a little while. If you have a really specific project in mind, then maybe some programming languages would be better, but if you just have general preferences you will never come to a consensus.
That said, Python is pretty flexible, it's the most popular on your list so the easiest to solve whatever sorts of problems you have by searching, so I'd recommend Python.
Considering the criteria listed below, which of Python, Groovy or Ruby would you use?

Criteria (Importance out of 10, 10 being most important)
Richness of API/libraries available (eg. maths, plotting, networking) (9)
Ability to embed in desktop (java/c++) applications (8)
Ease of deployment (8)
Ability to interface with DLLs/Shared Libraries (7)
Ability to generate GUIs (7)
Community/User support (6)
Portability (6)
Database manipulation (3)
Language/Semantics (2)