Hi fingermouse!
Well, a bone fracture needs a lot more than a week to heal and certainly a splint shouldn't be removed without a radiological check first. I suppose the fracture is stable and minor for the doctor to discharge you so easily. It can be healed without splinting but in that case one must be certain that no movement is allowed at the site of fracture. This means that you shouldn't use your finger at all but this is almost impossible without external stabilization (splinting). Moreover, one shouldn't risk a malpositioning during healing process by not having a splint.
My advice is to visit an orthopaedist (have the x-ray you already got with you), tell him/her your medical history and he/she should ask for another x-ray. Then he/she will guide you. In the meantime, avoid use of your hand or try to buddy strap your little finger with the ring one in order to be able to do some hand movement (immobilization is devastating and that's also why you should have the splint, so as to be able to move the rest of the hand).
Let me know of your progress.
ilias