Is it possible to save letters that are created through a merge as a document on the constituent gift entry? 

If I understand you correctly you are asking if you can link the actual word letters created in Mail as donor acknowledgment letters to the actual gift it is acknowledging.  Unfortunately no.  You can add the document as a media file on the constituent record but not to the gift.  I do not recommend adding them as Media files - it is very likely to get unwieldy.

If you are doing some other things in Mail (like follow up letters) those can be added directly to actions but nothing can be added to a gift to my knowledge

If you have purchased PaperSave you can attached word files to gifts.