In my experience you can't do much. If this happens, often the best thing is to just take a break from it all. It is just a page in an online encyclopedia. Okay many people read it - but sometimes you can't fix it.
Editors often "take ownership" of a page. And have decided on a particular way of doing things. There doesn't seem to be too much you can do, not if it is just two involved, them and you.
It probably would help to involve others though. If you can, in a civil polite way again. And if they are sympathetic.
Just do whatever you can while staying civil. You could try dispute resolution - but in my experience, that has a major risk of backfiring. Because the other editor is probably a long established editor with many friends, and whether or not, as an experienced editor, they know how to express themselves so that they will win these sorts of disputes.
Not just all the wikipedia policies they can invoke - but just how to handle the discussion so that they put themselves in a good light and you in a bad light for any third party reading the dispute and adjudicating it. I'm not saying deliberately necessarily - but that's how it works out.
Unless you have a lot of experience of how to handle them - or you have a really clear case, I'd steer away from dispute resolution.
Also on talk pages - they manage to steer the talk page discussions so that you are the one defending yourself, and so that you seem to be in the wrong and somehow always wrong footed, even though you know that what you are saying is correct. Whether deliberately or just how it turns out I don't know.
So - I tend to get dragged into these things, I'm not really the one to advise in a way - I am not good at handling them myself, easily caught up in them - but it really doesn't help at all.
But this is what I try to do nowadays - when I remember that is... .
Just make your points civilly in the talk pages. Don't say too much. Hope that others support you. Encourage others who suggest the same changes as you.
And - if that doesn't work - well - I'd walk away and find something more productive to do, such as answering questions in quora :).
Or indeed other parts of wikipedia. It is not at all a single thing. Where you are unwelcome in some pages, in others you may be welcomed by editors who value your expertise in your particular area.
If Wikipedia was a more friendly place I'd probably have got a lot more involved in improving Wikipedia. As it is, there are several pages that are in major need of improvement, that I keep away from because of combative editors. Can't do anything sadly. Is just how it goes.
Then there are other parts of wikipedia where I've felt I'm treated as a valued contributor.
So - it had its plus side though, ended up writing lots of answers here instead, as well as doing my own Science20 blog, which is much more interesting and fun.
But - long term Wikipedia - not always - but often tends to sort itself out.
Areas of wikipedia that were really poor a few years ago are now much better. These pages that I'd like to fix and can't - I don't think they are likely to be fixed this year or next year, but say five or ten years from now they may get fixed, new editors, old one stops editing, or loses interest or has a change of heart or new understanding of the subject, or whatever.
So you can console yourself with that - you've had your go at improving it, it hasn't worked, but - that doesn't annul your effort, which was well worth trying. And then maybe later someone else will be able to fix what you couldn't fix.
And meanwhile - if it doesn't resolve itself that is - maybe you can help fix other parts of wikipedia, and your efforts are probably more usefully engaged in that, chances are, if you want to help there.
And - another tip - if you edit a high profile page especially - likely to get a lot of resistance, chances are that if there is something you see as wrong with it, that - it might be that there are already people who have had previous battles about it that you have no idea about. You might be just the last in a string of editors that the resident editor, or a group of editors, has been defending that page against for years.
To take an example, there is a big ongoing RfC - really an all out battle but - quite civil, still - it's been going on for a long time - about whether to use the phrase "human-like" in the introductory paragraph of the article on Artificial Intelligence.
I'm not involved in that particular debate myself - made one comment to one of the RfCs - but there has been a whole series of RfCs and the last I looked, it seemed unlikely to resolve itself any time soon. Several points of view there, and they are finding it difficult to find a consensus. About what to outsiders seems a rather minor point probably - exact phrasing of the first sentence of the article.
Sometimes you get the same kind of thing going on where it seems really clear cut - that one of the editors is just out of date or mistaken in his views. Still, as there is no peer review or editor, then you can't always resolve that sort of thing either. You can try dispute resolution but that's just about editorial conduct, and has to be pretty major breaches of conduct also. You can try requesting expert assistance - and third party review - or a request for comment - but all of those work best in situations where you both agree that it needs resolution, not easy to do them by yourself.
And what's more the third party wikipedia editor reviewers you get - depending on the subject - often don't know much about the subject either. If you are trying to correct something that is popularly believed, but which researchers in the field itself know is incorrect - well third party reviewers on Wikipedia are likely to support the popular view on the matter. And they don't tend to read the citations if you offer citations to the original research. Well that's been my experience, rather limited, you may get better results.
But - instead you can try editing a more specialist page - with few editors. Maybe not much interest in it at all.
So then - out of the limelight of the big battle grounds of the main pages - you can sometimes help sort out the more specialist pages - and then leave the big main pages to catch up later on in their own time.
Or - just restrict your work on wikipedia for a while to help fixing typos and broken references and adding citations etc. Relaxing, easy to do, non controversial (usually).