If the character is woven into the action, it becomes mine to continue or tease out as needed.
If the character is not part of the action (like a story is done, we're in downtime), the departing player should 100% get to decide if his character sticks around, but I 100% get to decide if I want to deal with it, or just write them out. I generally don't love to NPC PCs, even former PCs, and most of the time, they fade away.
RL Examples
K wanted a new PC, and we wrote her PC going off on a personal quest. Later, I asked permission, and received it, to bring her back as an NPC, now taking leadership or their Mercenary company. She's now a Major NPC in the tale.
K (Same initial, different person) bailed with no notice mid-scene. I used his last few interactions to build a model in my head, and RPed his character also bailing. It might sound like a jerk move, but that's where the RP was leading. Made me wonder how sudden the departure was.
H had an RL thing come out to take her out for the foreseeable... In game, H finished out the battle, then had a thing come up to take her out of play.
A person I don't even remember the initial of ghosted mid-session. He was like ten sessions in, but I still just had his PCs ghost at the first opportunity.
I guess... I come down on the side of the Player when the player leaves room for it. Otherwise, I come down on the side of the story.