The spell check features is something that we have looked at in the past. However it looks like the http://wiki.languagetool.org project has gained some traction and might be work a look as it seems to be a nice fit.
I'll need to run sun some test to see how viable their approach for integration with our AcroForm and Annotation components is. I'll get back you about what I find.
At first glance https://www.languagetool.org seem to be a nice fit. The client example is pretty straight forward and uses a wrapper class to add spell and grammar check using red and blue underlines. Right click support for suggested fixes is also there. It's pretty impressive actually.
From an Acroforms and free text annotations point of view I'd have to do some integration development as we do quite a bit of manipulation to scale the widgets for different zoom levels. However this shouldn't be an issue for popup annotations as they are always drawn at the same size.
The library is LGPL 2.1+ and is quite large, with ~50 MB of supporting jars. The language dictionaries vary in size but there is quite a bit of sharing between similar regions. For example EN weights in around 5 MB with sub languages like UK or CA being about 500 KB.