What is the difference between fall off his bike and fall from his bike and which one of them is the best choice? to use in this sentence?
He fell off his bike and broke his leg.
He fell from his bike and broke his leg.
Both sentences are not only correct grammatically but also convey the same sense. The only diiference, I think, is that the use of "off" with fall/drop is more common in spoken English while the use of "from" with these verbs, though also used in spoken English, is a bit formal.
Those are basically the same, and I wouldn't pause if you said either one.
That being said, when talking about a bike I would naturally use 'off' rather than 'from'. I would be more likely to use 'from' if I was inside something, rather than on something.
For instance, I would fall from a car or from a plane, but I would fall off a bike, or off a surfboard. You don't get to use 'off' if you're inside something, but you could use 'out', which you can't use when you're on something. If you're on something but it sort of wraps around you, like a chair, then all of these are options.
I fell from a plane.
I fell out of a plane.
I fell from a skateboard.
I fell off a skateboard.
I fell from my chair.
I fell out of my chair.
I fell off my chair.
There's no discernibly blatant issue with using with either of them; they're both grammatically correct, but there are some small points to mention. The first is that fall/fell from his bike, to me (being a native speaker) implies a controlled descent from the platform.
John sat atop the wall, and peered from his perch, surveying the area for his bike. When he saw the item, he swiftly fell from the ledge to retrieve it.
On the other hand, fall/fell off his bike implies (again, to me) that the fall was part of an accident, or it wasn't meant to happen.
Upon going about the task of retrieving his bike, John fell off the wall and landed in a heap.
That being said, they are mere opinions of connotations, and should be taken with a pinch of salt. One last mention is that fell from sounds like more sophisticated language, so you shouldn't go out of your way to use it in colloquial speech or casual conversation.