By default there is no click or events that attached with any of the Views in react native like react.js. There are mainly two Views specifically designed for handling touch events. They are `TouchableOpacity` and `TouchableHighlight`. You can resolve your issue by wrapping your `Image` component with `TouchableOpacity` or `TouchableHighlight`.
<TouchableHighlight onPress={() => this.moveToAddNewCustomer()}> <Image style={styles.imagestyle} source={require('./ic_action_name.png')} /> </TouchableHighlight>
https://reactnative.dev/docs/touchablehighlight
https://reactnative.dev/docs/touchableopacity

You need to do it like this:
<TouchableHighlight onPress={() => this.moveToAddNewCustomer()}> <Image style={styles.imagestyle} source={require('./ic_action_name.png')} /> </TouchableHighlight>
or
<TouchableOpacity onPress={()=>this.moveToAddNewCustomer()}> <Image style={styles.imagestyle} source={require('./ic_action_name.png')} /> </TouchableOpacity>