import BottomActionMenu from 'react-native-bottom-action-menu';

//...

//To open
this.bottomActionMenu.show();

//To close
this.bottomActionMenu.close();

//...

<BottomActionMenu
  ref={bottomActionMenu => {this.bottomActionMenu = bottomActionMenu}}
  onClose={() => { console.log("Closed") }}
>
  {...}
</BottomActionMenu>

import React from 'react';
import { StyleSheet, Text, View, TouchableOpacity } from 'react-native';

import BottomActionMenu from 'react-native-bottom-action-menu';

export default class App extends React.Component {
  constructor(props){
    super(props);
  }

  showMenu() {
    this.bottomActionMenu.show();
  }

  action() {
    console.log('Any action here');
    this.bottomActionMenu.close();
  }

  render() {
    return (
      <View style={styles.container}>
        <TouchableOpacity onPress={() => { this.showMenu() }}>
          <Text>Open menu</Text>
        </TouchableOpacity>

        <BottomActionMenu
          ref={bottomActionMenu => {this.bottomActionMenu = bottomActionMenu}}
          onClose={() => { console.log("Some action to take during closing") }}
        >
          <Text style={styles.title}>An example of a title!</Text>
          <Text style={styles.text}>
            Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat.
          </Text>
          <TouchableOpacity style={styles.btn} onPress={() => { this.action() }}>
            <Text>OK</Text>
          </TouchableOpacity>
        </BottomActionMenu>

      </View>
    );
  }
}

const styles = StyleSheet.create({
  container: {
    flex: 1,
    backgroundColor: '#fff',
    alignItems: 'center',
    justifyContent: 'center',
  },
  title: {
    textAlign: 'center',
    fontSize: 17,
  },
  text: {
    textAlign: 'justify',
    marginTop: 20,
  },
  btn: {
    backgroundColor: '#CCC',
    height: 40,
    borderRadius: 10,
    justifyContent: 'center',
    alignItems: 'center',
    marginTop: 40
  }
});