ScrollView inside an PanResponder

Try using a function for onMoveShouldSetResponder so it only swipes horizontally when gestureState.dx is far greater than gestureState.dy like so:

onMoveShouldSetResponder: (evt, gestureState) => {
    return Math.abs(gestureState.dx) > Math.abs(gestureState.dy * 3);
},

You could also have a function in onPanResponderMove tracks the direction of the swipe gesture and then reset in onPanResponderRelease so you don't have problems when a vertical swipe changes into a horizontal swipe like so:

checkSwipeDirection(gestureState) {
    if( 
        (Math.abs(gestureState.dx) > Math.abs(gestureState.dy * 3) ) &&
        (Math.abs(gestureState.vx) > Math.abs(gestureState.vy * 3) )
    ) {
        this._swipeDirection = "horizontal";
    } else {
        this._swipeDirection = "vertical";
    }
}
canMove() {
    if(this._swipeDirection === "horizontal") {
        return true;
    } else {
        return false;
    }
}

Then use it like so:

onMoveShouldSetPanResponder: this.canMove,
onPanResponderMove: (evt, gestureState) => {
    if(!this._swipeDirection) this.checkSwipeDirection(gestureState);

// Your other code here
},
onPanResponderRelease: (evt, gestureState) => {
    this._swipeDirection = null;
}

I found an awesome article online by Satyajit Sahoo on Medium How I built React Native Tab View.It shows more in-depth how to implement your own Tab View. I recommend taking a look at the blog post as it was really helpful for me.

Update: Check out the documentation here Gesture Responder Lifecycle if you want a parent component to prevent a child component from becoming the gesture responder or vice versa.