I don't think you know what a launcher does.
You're asking a question that implies that you think the launcher propels the missile, like a bullet from a gun. It does not. Missile launchers are basically just tubes to contain the blast, plus electronics to feed info to the missile, and some sort of trigger to tell the missile when to go.*
You can very easily have a simple, disposable tube that protects your mini-missile and has a button to trigger it. In fact, that's basically what a LAW is, and it is tremendously popular with users. For something meant to engage smaller-scale targets, you could scale that down pretty easily.
But can you skip the tube entirely? Kind of, it's just a bad idea. Clearly you can; that's exactly how air-launched missiles already work. You could move the arming electronics onto the missile, and beef it up to not need protection in transport, and probably even come up with some way of making it air launched after throwing it. But you end up with something bulkier, far less accurate, and with a smaller payload. Also, it probably requires MUCH more training to use effectively.
All in all: yes, but it's worse.
*There are a few exceptions. For example, submarine launchers usually use compressed air to throw the missile clear of the surface. Similarly, recoilless rifles can be man-scale and they look a lot like missile launchers and do throw a projectile. But the kind of self-propelled missile you're contemplating is usually just sitting in a plain tube.