How can I put an icon inside a TextInput in React Native?

You can use combination of View, Icon and TextInput like so:

<View style={styles.searchSection}>
    <Icon style={styles.searchIcon} name="ios-search" size={20} color="#000"/>
    <TextInput
        style={styles.input}
        placeholder="User Nickname"
        onChangeText={(searchString) => {this.setState({searchString})}}
        underlineColorAndroid="transparent"
    />
</View>

and use flex-direction for styling

searchSection: {
    flex: 1,
    flexDirection: 'row',
    justifyContent: 'center',
    alignItems: 'center',
    backgroundColor: '#fff',
},
searchIcon: {
    padding: 10,
},
input: {
    flex: 1,
    paddingTop: 10,
    paddingRight: 10,
    paddingBottom: 10,
    paddingLeft: 0,
    backgroundColor: '#fff',
    color: '#424242',
},

Icons were taken from “react-native-vector-icons”