react-native - Fit Image in containing View, not the whole screen size

Set the dimensions to the View and make sure your Image is styled with height and width set to 'undefined' like the example below :

    <View style={{width: 10, height:10 }} >
      <Image style= {{flex:1 , width: undefined, height: undefined}}    
       source={require('../yourfolder/yourimage')}
        />
    </View>

This will make sure your image scales and fits perfectly into your view.

If you know the aspect ratio for example, if your image is square you can set either the height or the width to fill the container and get the other to be set by the aspectRatio property

Here is the style if you want the height be set automatically:

{
    width: '100%',
    height: undefined,
    aspectRatio: 1,
}

Note: height must be undefined

Edit (Based on @rob-art's comment):
If your image is a different aspect ratio than the one you want to set in the style you can use resizeMode to control how the image should be displayed. Use resizeMode:'contain' to ensure your image is not cropped.
See documentation for more details

Anyone over here who wants his image to fit in full screen without any crop (in both portrait and landscape mode), use this:

image: {
    flex: 1,
    width: '100%',
    height: '100%',
    resizeMode: 'contain',
},