First, Kudos for picking an article that is actually linked to the base results they mention. Even so, no scientific study goes without criticism and critique. I read the article, and also the original reference. It is very short, more a mention than anything else.
A couple of points. It looked at death rates of women who birthed from 1964 and 1991, in Denmark. While Denmark is/was an advanced country medically, the changes that occured in reproductive are during those 3 decades is phenomenal.
EDIT: Still trying to track this, but Wikki apparently says that DK only legalized abortion, except in cases of rape/incest and medical need in the 1970's.
ALSO, even in the US, all statistics I have been finding so far clarify that safe and legal abortion is safer. However, illegal abortion still happens, and is on the rise with the increase of highly restrictive laws in some areas.The study makes no causal connection at all. It simply says that more than a year after birth or abortion or miscarriage does the women die. It could well be that women in riskier professions or engaging in riskier lifestyles (be they prostitutes.. an essentially legal profession in DK, in the military -- or other), along with women who are experiencing medical issues being more likely to abort.
A hint a what Woodruff said -- that women were more likely to abort fetus when there were issues with her or the child because there was ALSO a blip for miscarriages. That is women who had abortions were more likely to die, but women who had miscarriages were ALSO more likely to die. Let me add that there is no real stigma with out of wedlock births in Denmark, nor a problem attaining help caring for the child and has not been for quite some time. So, there is much less pressure to abort a child for "social reasons" than there is here in the US. In other words, its likely that women in Denmark are more likely to choose abortion for purely medical reasons than women here in the US. This would tend to slant the data toward a higher mortality rate.
None of this can be answered because the bit is so short and lacks explanation.
Another point is that this data does go against, not, as the article you linked to indicates, "commonly accepted" ideas. It goes against STUDIED and PROVEN data in other countries across a wide range of circumstances. The study cited seeks to look at death rates occuring more than one year down the road.. but is that really meaningful in reference to abortion/birth? Complications are most likely to cause death shortly after the procedure, not a year down the road, which is why most studies have not carried the research out that far.