Spell check is working just fine in HTML Viewer here. Of course that’s because Mac has a global, built in spell check.
See it work in anything using HTML Edit on a Mac.
Safari 9.0.1 on Mac OS 10.9.5
Try out the HTML Edit demo or Answers and see if spell check works for you.
If it doesn’t then there is something on your computer making it not work, or mine making it work (which we can figure out later.) However, if spell check works in the demo then it’s your HTML Viewer setup.
I went to go confirm it works on 10.10 and discovered what may be the reason it’s not working for you.
You must enable spell check in the area with right-click > spelling > check spelling while typing
This was off on my 10.10 install, but since turning it on in that instance spell check works now.