First reaction: "That just sounds like murder with extra steps."
Imagine a culture [...] where if you want a divorce, you have to fight your spouse to [the] death in a public duel.
First of all, you'll need some powerful third party (such as the Church) to say "We regulate who's allowed to get a divorce, and we say you can't get a divorce unless you kill your partner in a public duel." Historically of course there's no problem with this; the Church did (and in many cases still does) regulate divorces.
But wait, what does it even mean to "get a divorce" from a dead person? In our universe, AFAIK, the Church permits widow(er)s to remarry. If your universe is similar, then anyone wanting a divorce won't have to engage in a public duel if they don't want to: if they want to stop-being-married badly enough to actually kill their spouse in public, then certainly they want it badly enough to kill their spouse in private. Alice could fight Carol in a public duel to the death, or Alice could just put some strychnine in Carol's tea. The end result is the same, right?
...Unless of course your judicial system also imposes worse penalties for regular murder than for death-by-judicial-duel, which I'm sure it ought to. Then Alice might prefer to run the risk of a duel (where there's just a one-time risk of death-from-Carol), rather than attempt a murder (where the risk of detection and punishment never goes away, and has the manpower of the state behind it).
So that's the calculus from Alice's point of view.
The main problem with this idea, to me, is that a "duel to the death" really only works if you have two parties who are both willing either to kill, or to die.
Suppose Alice wants a divorce and Carol doesn't — Carol wants to stay married to Alice. In that case, Carol certainly doesn't want to kill Alice; she wants to keep Alice alive so they can stay married. (Maybe get some couples therapy or antidepressants, maybe just lock her in the basement, but not kill her.) If the duel's only possible outcomes are "Carol dies and the marriage is dissolved" or "Alice dies and the marriage is dissolved," then none of the outcomes are going to be satisfactory to Carol.
Suppose Alice wants a divorce and Carol also wants a divorce. They have already agreed that a divorce is in their best interests, and want to part amicably. But the Church (or whoever your third-party power is) won't let them; it insists that they can't divorce until one or the other of them is dead. (Again, this is the historical position of the Catholic Church on divorce, so it's quite realistic even if it doesn't seem very sensible by modern standards.)
In the latter case, we simply have "the Church disallows divorce" with extra steps. In the former case, your Church will have to decide which party has more power here: Alice, who wants the divorce badly enough to duel for it? or Carol, who wants the status quo? Can Alice force Carol to duel? Or can Carol force Alice not to duel?
In any case, if duel-to-the-death is the only way to get a divorce, I think the ramifications will be pretty much the same as the historical "no divorce ever" policy.
But there's plenty of room for legalistic loopholes here! Is there some way for Alice and/or Carol to exploit the public-duel law to get a divorce while both of them remain alive?
Bribe the examining physician to declare Alice dead. The divorce is granted, and then when Alice mysteriously comes back to life a week or a minute later, some form of "double jeopardy" kicks in and the two remain divorced although alive.
Have Alice and Carol both nominate champions to fight in their place. Whichever champion dies doesn't really matter; the divorce is granted either way. This reduces the law from its original moral purpose to what I would call a "circus tax" on divorce. Want a divorce? Okay, but you have to sponsor an expensive gladiator duel for the public benefit.
Have Alice nominate a "champion" to fight in her place, but make it someone expendable, like a household slave. Carol kills the slave and the divorce is granted.
This third option actually seems the most dramatically interesting to me. (Recall that Alice is the one who wants the divorce; Carol is the one Alice is trying to escape from.) Alice must want out badly enough to send a slave to certain death. Carol is forced by Alice to commit what could feel subjectively like murder. Alternatively, Carol could display too much ruthlessness in the slaughter, thus (in the court of public opinion) proving that Alice was right to want to escape. Either way, stigma attaches to Carol... or to Alice? Both roles seem distasteful to the modern reader. So you, the writer, have the freedom to decide what your society thinks about this, and either way might end up surprising the reader with what happens.
(Assume a spherical marriage: I used "Alice and Carol" throughout in order to sidestep the implication of one party being physically stronger than the other, and to simplify the case analysis. With a heterosexual marriage, you'd have to consider whether to apply the same legal rules for "Alice wants a divorce and Bob doesn't" as for "Bob wants a divorce and Alice doesn't.")