The Duel for Catelyn's Hand

[Pod The Impaler]

It is not a trial by battle, gods aren't involved.

It is inherent to settling a matter by single combat that one is asking the gods to render a decision concerning that matter.

No, without official sanction, it is technically just two boys fighting over a chick. But Brandon Stark, if challenged, is not the sort of guy to turn down a fight. Baelish was a puny nobody, but he'd have probably fought that duel even if Jaime Lannister was the opponent.

There's no evidence Hoster even knew about the duel before it took place, much less approved of it. (Really, why would he allow it? One of them would almost certainly die, and that would be trouble for him.)

Still, he certainly had to deal with the messy aftermath of that, and Baelish of knocking up his other daughter.