Is it possible to say 'I've broken my leg, I can't be walking' regarding it's happening at the moment or I must always say 'I can't walk'?
In most contexts, I can't walk will be a simple statement of fact. I have broken my leg - I can't walk means that I am incapable of walking; it is impossible for me to walk.
The can't + be + present participle is a less common but grammatical construction that usually conveys not so much impossibility as undesirability. Here are some Google examples that indicate the speaker is indeed capable of walking, but considers it undesirable:
I can't be walking around like a sleep-deprived grump.
I can't be walking around looking like a junkie.
I can't be walking around with a sourpuss on my face
I can't be walking around the hood with no Disney ass name!
I can't be walking around with that price tag on my head.
The progressive form is used to emphasise the ongoing nature of the walking in such contexts.
So now to the question: "Is it possible to say 'I've broken my leg; I can't be walking' regarding it's happening at the moment or must I always say 'I can't walk'?"
As always, the context is decisive in answering questions about choosing between two verb constructions. In the context of someone calling you to see if you'd like to go shopping (and not knowing about your injury), the most likely response is I've broken my leg - I can't walk" (incapability).
If, on the other hand, you only have a slight fracture and the doctor has advised you to stay off your feet, then you could say: "I've broken my leg - I can't be walking" (undesirability).