Can one "marry one's wife"?

I think "he married his wife" is merely redundant, not illogical. (However, in terms of language, bear in mind that it need not be proven logical in order to be considered grammatical English and to be understood by everyone.)

I think it is quite normal to have an attribution (e.g. relationship, title, name, etc.) be understood to be the person that fits that attribution at the time of the utterance or writing. Now, this isn't always done, but I think it is the default assumption, and it is also logically consistent.

This is why, in news articles, we see things like, "Sean Penn and then-wife Madonna were often seen at...". If the referent of "wife" is dependent on the tense or timeframe of the sentence it is in, then there would be no need to say "then-wife", you could just say "wife".

Think of other examples where this is done every day:

President Obama attended Occidental College.

But he wasn't president when he was in college.

My father was in the army for a few years.

But he wasn't my father before I was born.

And so on. I think that once you start thinking about it, you will see it is actually done all the time, and you probably didn't even notice.

"Married" is past tense, obviously. We are looking backwards in time from now to see the action that did occur. However "He" and "his wife" are the current topic of conversation (as of the day of the obituary). As of now, being after the occasion of the past-tense verb, they are "he" and "his wife". So yes. this is valid.

Had it been the day before they married, it would merely be poetic license, as they would not, yet, have been married.

It's simply shorthand for saying "He married the woman who is now his wife."

This is more of an extension to the other answers.

The verb "marry" can be used with reference to others as well. A minister "marries" couples. This leads to legitimate, but improbable, uses of the word.

Of course, if you "married your wife" in this sense, that means that you married a couple consisting of your current wife and someone else. They got divorced/whatever and then you "married" her later (in the sense that she became your wife). Pretty improbable use-case.

"Marry one's wife" is even stranger. This is only possible when "marry" is in the future tense--"I will marry my wife tomorrow", and it means "I shall marry my wife to someone else soon(after getting a divorce)". Even more improbable, since I doubt someone would say that.

So, if you want to be nitpicky about logical impossibilities, then the above two are some cases where the phrase can be used correctly. Extremely improbable use-case, though, even if you consider bigamy and the likes.