Let's say a Supreme Court Justice resigned today because of health reasons or whatever...

I think you have to be part of the court when the order is issued.

He won't resign. Ever.

He was only allowed to sit and be silent. Obviously.

But not, afaik, at the Supreme Court.

I don't know if this is policy of if the question has ever come up.

That's when the abortion and NY gun-control cases will be decided. I expect a 5-4 decision on the former, and a 6-3 one on the latter.

And a purely hypothetical answer would be: eight justices will make the decision. Not sure what happens in the event of a tie. I believe the case goes back to the previous court or the decision of that court prevails.

But I very much doubt that Kavanaugh will resign any time soon.

You can't vote on a case before hearing it, so he couldn't vote. They had only eight the entire time they left a seat vacant because it was "an election year." Still hard to believe they were able to get away with that.

If they haven't voted, then the court votes with whatever members are left.

Then the decisions gets committed in writing by whomever is assigned to writing the opinion, in a process that can take awhile.

So, if it's after oral arguments/conference for a case that the hypothetical justice departs, their vote would normally count. If they departed before arguments on a case, they are no longer involved in the outcome.