No, the sentence is not correct in modern idiomatic English.
When in doubt, simplify the sentence. This type of error only ever occurs in complex, overconstructed sentences, where less fluent speakers (including native speakers) can be tricked into forgetting simple things like agreement and conjugation. In this case, we simplify:
When all past societies had separately prospered in peace within the local borders of their own solutions, now that they are melting in one big pot an impartial vision is due, whereby common grounds and differences be suitably understood.
Now that they are melting, an impartial vision is due, whereby common grounds and differences be suitably understood.
Now a vision is due, whereby grounds and differences be understood.
Now a vision is due, whereby things be understood.
Things be understood.
Is "Things be understood" idiomatic in modern English? No--although it would have been in some archaic dialects, and on Talk Like A Pirate Day ("Aye, it be so").
The sentence could be corrected in many ways, while maintaining what was presumably the intended meaning:
...whereby grounds and differences can be understood.
...whereby grounds and differences are understood.
...whereby grounds and differences may be understood.