A date picker on iOS is as simple as adding the DatePickerIOS component. On Android, it's more complicated and requires a specific API; that's why it's usage is covered in React Native official API docs:
https://facebook.github.io/react-native/docs/datepickerandroid.html.
For iOS, though, we need to pass date prop for the currently selected date and onDateChange function to update our state when the date gets updated as a result of user selection. We also can pass minimumDate and maximumDate to define our date range and timeZoneOffsetInMinutes to define our timezone. The default timezone will be our device's timezone:
<DatePickerIOS
date={this.state.date}
onDateChange={(date) => this.setState({ date })}
/>
You can read more about all these props in official docs:
https://facebook.github.io/react-native/docs/datepickerios.html