Not at all. Only desktop browsers do and only a spellcheck (no grammar analysis, which is available only with plugins, however, plugins are buggy for custom input fields (like Discourse) or probably not user-friendly (I use two of them)). Any mobile browser under Android I have tested (Chrome, Firefox) do not have any spell check and will probably not in any upcoming feature.
Is it possible to add a plugin that will interact with a local or remote server via post and get JSON API to provide correction for user input in Discourse? It will need to mark errors and provide suggestions for corrections, e.g. popup balloon on mouse hover over the marked place, where you can click to select the correct variant of spelling or grammar.