App.jsx

export class App extends React.PureComponent {
    render() {
        return (
            <View style={{width: '100%', height: '100%'}}>
                <OtherComponent />
                <SpecialComponent ref={(ref) => ComponentHelper.setReference(ref)}/>
            </View>
        )
    }
}
// ...
export class OtherComponent extends React.PureComponent {
    render() {
        return (
           <TouchableOpacity onPress={() => ComponentHelper.show(300)}/>
        )
    }
}

// ...
export class ComponentHelper {
    static _comp;
    static setReference(element) {
        this._comp = element;
    }
    static show() {
        return this._comp.show();
    }
}