Conflict of interest: My research reflects badly on my PhD supervisor
I had interesting situations during the final stages of my PhD career, and I found that my dissertation committee was an incredible resource. If you are in the final stages of writing your thesis, that means you have a dissertation committee (or have otherwise established a collegiate network of professional academics that support your PhD training). Now is the time to use this resource. Ask to speak to them in person about the problem. Specifically, you should approach your committee chair (who should be different than your supervisor). If your chair is your supervisor, then just speak to the most senior person in your committee. 

The way that I understand the power dynamics / vested interests of the whole PhD situation is that your supervisor has inherent conflicts of interest - they want you to do research that supports them. Thus, they 1) are better off keeping you as long as possible (but not too long) and 2) want your research to support their research. Your committee chair, on the other hand, wants to see you graduate. Your department chair also wants to see you graduate. Your supervisor wants to see you graduate, sure, but as I mentioned, they have other motives. 

Ultimately the goal is to keep everyone happy, and especially your supervisor. Having your PhD supervisor on your side is key to a successful career in Academia (or at least makes getting recommendation letters are lot easier). 

Use this situation as a learning experience (one of the many tangential learning experiences associated with a PhD) for how to navigate the people problem of science. So in conclusion- 

1.) Give your supervisor the benefit of the doubt. Try to understand their points and try to actually discuss it with them - IN PERSON. Abandon email for this interaction - you need the full spectrum of our human communication abilities. 

2.) If #1 fails, engage your dissertation committee. This exact situation is a fine example of why dissertation committees exist.

3.) If #2 fails (or you're in a program that doesn't have dissertation committees), engage another trusted senior investigator if you've created relationships with other investigators.

4.) If you get to here, go to the department.