What should I do after my children (13-year-old and 17-year-old) have had a big fight and are now not talking to each other for over a whole day?
It's pretty natural not to talk after a big fight; if not bound to one house, they would probably go their separate ways. How I would react depends to some degree on what the fight was about, whether hurtful things were said that were untrue, etc.
If you try to fix it immediately, you risk the kids feeling resentful that you "just don't get it". But you can ease the tension (and show them you love them and care about their relationship) by individually speaking to each child, supporting them where you can, not invalidating their feelings (they feel what they feel), and offering them gentle guidance. These things may well cut down on the duration of the tension. It is extremely helpful to model this kind of behavior as an adult as well.
I tend to be someone who respects feelings. It's pretty unnatural not to have tension after a big fight. If it was over something trivial, I would let my kids calm down with their own thoughts and work it out themselves. If the fight was a one-sided violation of the other child's rights, or I saw something harmful going on (grudge holding, provocative comments, etc.), I would get involved in that, first by speaking with each child individually and listening carefully, thinking about it, then getting the two of them together and moderate a discussion and encouraging some problem solving techniques. I would do this with "children" of any age.
I guess what I'm saying is that whether or not I got involved to a significant degree depends on the fight. One thing I used to do was to make my kids apologize for their part in a significant fight. That has backfired with one of my kids, who doesn't apologize now even when he's wrong. I don't know how much of that is my fault and how much is just him (he was my most difficult child.) It saddens me, though, and I wonder if I should have done things differently.
Parents should examine their motives before interfering in the responses of children. Do you want them to speak to each other because you think it would be better for them or for you? Certainly, you would feel better if they could be more "mature", but do you talk to someone right after a big fight? Sometimes it's best to gather your thoughts before doing so. But it hurts to see your kids hurting, so it's understandable that you want to "fix it". But determine who benefits from fixing a rift; that will determine your approach.