How to make PCA9685 work with Arduino Due?
I made it working by this way: despite PCA9685 datasheet recommendations, we do not need to connect OE pin to VCC. Also we do not need to connect OE to ground as it recommended in some instructions related to other Arduinos.

So we should leave OE pin unconnected.
Also we should use 4,7K pull-up resistors to connect SDA and SCL to VCC.

The board can be used without level converters.

No special program/libraries changes needed. So it was a hardware problem.