Extending native spell check
The system-wide spell-ckecker is accessed through System Prefs > Keyboard > Text... top right, 'Correct spelling automatically'...
 but I'm not sure Chrome doesn't just ignore it, it does most other things set system-wide. 

If you look in the Edit menu in Safari, for instance, you'll see "Spelling and Grammar" &amp; also "Substitutions"
Chrome only has part of the Spelling and Grammar menu, &amp; Substitutions is missing entirely.

So, I think the TL:DR is Chrome can't/won't do this.