I've been the user leaving comments before. There are a couple of things worth bearing in mind.
First off, the user commenting in all likelihood doesn't realize that their comments are coming across as curt. This often happens when there is a large amount of these low-quality posts. It's not really sustainable to write a personalized comment for a hundred different answers, so one thing that users do is to create a stock comment, and post that with minimal edits on each answer. So you've got this user leaving this stock comment, that they don't realize is curt, on a bunch of different posts.
Then, you've got the user on the receiving end of the comment. They don't know that it's a stock comment, so they've just gotten a comment pointing out that their answer doesn't meet requirements, which stings in the first place, and the comment is curt, to boot.
An overworked, tired user and a frustrated new contributor. Not a good combination.
Of course, neither of these people actually have an excuse, so you do need to deal with the situation. What's the best way to approach it?
If things are ongoing, the first step you need to take is to stop the immediate situation. If it's happening in chat, or comments, and things are degrading, you need to stop the immediate situation - essentially separating the users. This could be achieved by simply deleting the comments, or locking the post, or freezing a chatroom, or trashing messages, depending on what seems like the best option at the time. Try to keep it impersonal, and consider telling them that you'll reach out to each of them personally.
Second off, reaching out to the offended new users would be a good next step. Apologize for the curtness of the previous commenter. Acknowledging that there was a misstep is always a good idea. Once that's done, you can explain to them that even if the other person was curt, being out of line in response is still not a appropriate response. If it seems like it could end productively, work with the user to help them improve their answer or whatever.
In addition, reach out to the commenter. Explain that their comments have been curt and could be improved, and that they need to improve it a bit. Thank them for the effort they've been putting in. Including praise and gratitude is a great way to get them to be more receptive to what you're saying - just telling someone off is unlikely to be received well. Make sure to balance appreciating what they're doing but informing them that they need to improve things a bit.
Of course, things will be dependent on the exact situation. In your position as an exception handler, you'll need to make a judgement call about the specific circumstances you find yourself in and what steps need to be changed.