Once again, Naoufal Kadhom, has released a library to enhance React Native's features.

react-native-activity-view on GitHub

react-native-activity-view is a React Native library for displaying iOS share and action sheets.

var ActivityView = require('react-native-activity-view');

var YourComponent = React.createClass({
  _pressHandler() {
    ActivityView.show({
      text: 'ActivityView for React Native',
      url: 'https://github.com/naoufal/react-native-activity-view',
      imageUrl: 'https://facebook.github.io/react/img/logo_og.png'
    });
  },

  render() {
    return (
      <View>
        ...
        <TouchableHighlight
          onPress={this._pressHandler}
        />
          <Text>
            Share with Activity View
          </Text>
        </TouchableHighlight>
      </View>
    );
  }
});