I posted another solution here Which is the best approach to insert an vector (svg) graph into a react native application, This approach use a vector font(from svg) instead a svg file. PS: Its works great in ios and android, and you also can change the color and size of your vector icon.
If you want to insert a svg directly into your app, you can try a 3rd party library : react-native-svg. With more than 3k stars in github it's one of the best approach.
And here's a sample:
import * as React from 'react';
import SvgUri from 'react-native-svg-uri';
import testSvg from './test.svg';
export default () => (
<SvgUri
width="200"
height="200"
svgXmlData={testSvg}
/>
);