App.js

import React, { Component } from 'react';

import { AppRegistry, StyleSheet, View, Platform, Picker, ActivityIndicator, Button, Alert} from 'react-native';

export default class App extends Component {

 constructor(props)
 {

   super(props);

   this.state = {

   isLoading: true,

   PickerValueHolder : ''

  }
 }

 componentDidMount() {

      return fetch('http://provareact.altervista.org/reactNative.php')
        .then((response) => response.json())
        .then((responseJson) => {
          this.setState({
            isLoading: false,
            dataSource: responseJson
          }, function() {
            // In this block you can do something with new state.
          });
        })
        .catch((error) => {
          console.error(error);
        });
    }

    GetPickerSelectedItemValue=()=>{

      Alert.alert(this.state.PickerValueHolder);

    }

 render() {

   if (this.state.isLoading) {
     return (
       <View style={{flex: 1, paddingTop: 20}}>
         <ActivityIndicator />
       </View>
     );
   }

   return (

    <View>

          <Picker
            selectedValue={this.state.PickerValueHolder}

            onValueChange={(itemValue, itemIndex) => this.setState({PickerValueHolder: itemValue})} >

            { this.state.dataSource.map((item, key)=>(
            <Picker.Item label={item.nome} value={item.nome} key={key} />)
            )}

          </Picker>
                <Text>Mellaucci</Text>
          <Button title="Clicca qui per vedere gli elementi del picker" onPress={ this.GetPickerSelectedItemValue } />

    </View>

   );
 }
}