One should be able to learn React Native directly through this tutorial, without starting with React for web. The main prerequisite is ES5 JavaScript. The tutorial makes analogies to HTML/CSS/React and other web technologies, but these aren't crucial to understanding the material.
Ideally web is just another taraget :) https://github.com/necolas/react-native-web
That said, while writing a website with react-native-web is potentially interesting for new sites, it can be less worthwhile for an existing site to just embed a little bit (which is one of the huge values of React). It's also still very experimental. But I suspect it will be supported in a huge way soon.
You're correct that there's no theoretical reason that RN apps can't run on the web; as dabbott mentions, react-native-web is one third-party attempt at this. We'd like to support this officially one day but don't currently have a high-quality solution we can recommend.