I know this is 6 years later, but I see people still struggling with this. Also there is a very simple answer to this question, there is no need for all those hacks posted above. The issue causing this is the value prop, once you remove the value prop your native behavior will be back.

Why? Because once we use value, this becomes a controlled TextInput meaning every keystroke triggers onChangeText and it becomes controlled by React. This can break the native behavior.

only having onChangeText and updating the state is uncontrolled manner, which is controlled by the native platform therefore allowing you to double tap for a period -- also increase performance since not everything rerenders on every keystroke. Do not use value unless you really need it.