What you are looking for are called Bottom Sheets in Material Design (see here for examples) and Action Sheets in iOS (customized action sheets more precisely, like this one).
I found two React Native libraries which implement Bottom Sheets (so only for android):
- https://github.com/cesardeazevedo/react-native-bottom-sheet-behavior
- https://github.com/WhatAKitty/react-native-bottom-sheet
And this library implements custom Action Sheets (so only for ios):
- https://github.com/eyaleizenberg/react-native-custom-action-sheet
I didn't find any RN library which works for both platforms, so we will have to be happy with what we've got. Good luck!
Edit: Since my answer a new package came out: https://github.com/beefe/react-native-actionsheet. Thanks @thibautnoah for pointing it out.