I would be crushed to get a negative review as well, especially if I felt I did an excellent job! I read the review and your response to it, and I think you handled it extremely well. You clearly but non-defensively asserted your disagreement and confusion, provided your perspective on the services you provided, and asked for specific feedback.
My main suggestion would be changing your response from passive to active tense. Instead of saying "regular exercise was provided," switch it to "I took Leonidas on regular walks" (or whatever exercise/play you did). It's a minor change, but it emphasizes the work that YOU did; it didn't just happen - you put in the effort. Also, if it's accurate and reflects positively on your services, I would be specific about how often you sent updates or were in contact with the owner. If you give them the benefit of the doubt, you could say "Maybe there was a technological glitch we weren't aware of, but I sent updates every day," but if they responded to the updates, you could very gently state that you sent regular updates and figured there weren't any glitches since they responded to your messages. Also, if it's true, I would explicitly say that you did not notice any flea problems or rash, etc while he was in your care since this is a major point that you did not directly address.
Another option, and this is just me thinking aloud not necessarily a recommendation, is to offer the owner some kind of amends or some effort to repair the customer-sitter relationship (if you want to do that). You could message them and offer an apology that they FELT you were inadequate and emphasize that that's certainly not what you intended. You can say you wish there was something you could to do (this is a gesture, not an offer - do NOT offer to pay for anything). This process would not be about admitting fault (since you may very well not be at fault at all), but about acknowledging that there was a difference in perspective or a misunderstanding, etc and informing them that this actually matters to you. Also, I would ask how their dog is doing now and if the pup is feeling better. This reminds them that you actually care. All of this may help to remind them that you certainly didn't mean to hurt their dog and they may soften their review some.
Your last option (as far as I can tell) is to contact Rover support to see if there's any way for them to support you. I'm not sure what they could do, but if you have a concern that a customer is being misleading or dishonest, they should at least be aware in case this happens with their next sitter.
If you're concerned about how this review will affect your ability to get new customers, I would suggest... (more)
I also got a bad review, nobody booked me since then. The most upsetting thing.