This is an interesting paper. All papers that look at what might be characterized as the "unintended consequences" of the ACA are interesting.
Of course, the data is only "suggestive" that Medicaid expansion reduced divorce rates in expansion states, since, as the authors point out, we can only surmise "medical divorce" in many situations where couples divorce after a major life-ending high touch care diagnosis for one (think Alzheimer's and related dementia). I suppose this is because most people do not disclose their reasons for divorce -- they are not required to under a no-fault divorce system -- and often don't disclose medical divorce at all (except in official records) because legal divorce and estate planning is still stigmatized, particularly among older people.
Medicaid Divorce has long been an option to consider for retirement planning community spouse income and asset preservation. Here is a pretty good article discussing why; http://www.huffingtonpost.com/rev-amy-ziettlow/is-divorce-the-best-option-for-older-americans_b_6878658.html. I move the discussion to Medicaid Divorce because Medicaid pays for at least half of the long term care in this country. (http://publish.illinois.edu/elderlawjournal/files/2015/08/Miller.pdf).
One of the things I find most interesting is that the article actually tries to take a stab at quantifying medical divorce. There has generally been little data on how common the practice actually is, even when insiders know that it is done. What is cool about this paper is that it gives us a number to consider. Given how little we have really known about the prevalence of the practice, even suggestive data (such as this) is quite interesting.
American divorce rates and marriage rates have always been influenced by economic factors (https://www.washingtonpost.com/news/wonk/wp/2015/06/23/144-years-of-marriage-and-divorce-in-the-united-states-in-one-chart/?utm_term=.47c76b606994) but this is quite a reduction over quite a short time interval, even though divorce rates are in decline (https://www.nytimes.com/2014/12/02/upshot/the-divorce-surge-is-over-but-the-myth-lives-on.html) overall.