If we're assuming that a missile has a similar sort of soft launch capability to the Javelin but with a smaller and lighter missile, could it be plausible to have a missile that is sort of hand thrown/dropped rather than requiring a dedicated launcher?
I'm thinking here of a setting with a loosely similar tech level to The Expanse. Part of the idea here is that missiles/gyrojet weapons are popular onboard spacecraft because of the lack of recoil.
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.
Your problem isn't the hand-thrown missile
Having goofed around with model rocketry during my youth, I can easily imagine a hand-thrown missile. The only real problem is when to ignite the fuel. But you could get around that problem with a number of ideas:
Once thrown, an initial puff of compressed air pushes the missile away from you before the fuel ignites (not dissimilar to a Submarine-Launched Ballistic Missile, which is launched using compressed air).
Hold/tie-off a string that, at extension, triggers the fuel.
An on-board altimeter chip senses a high-enough throw to be far enough away from the thrower.
So, can you launch said missile? I believe this is well beyond the realm of suspension-of-disbelief.
Your problem is aiming the darn things
I was once a micro-publisher and during that time I coined a phrase, "technology dichotomy." If your soldier has the means of "painting the target" then said soldier also has access to tech that renders the need to throw a missile entirely moot. Think of it this way: if your soldier is packing around enough tech to actually aim the missile, then there's nothing stopping your soldier from having, for example, a backpack-mounted launching system.
But maybe what you're looking for is something like a rocket-assisted hand grenade...
In a phrase... no you aren't. It sounds cool and who wouldn't want to lob a grenade over a mountain? AmIright? But if you think about it, any kind of assist to throwing a grenade makes the grenade less accurate. What's the point of lobbing a grenade a half-mile if you can't hit the broad side of a barn?
Which brings us back to painting the target...
Which brings us back to what's the point of hand-throwing the missile of you have better ways of solving the problem thanks to the tech needed just to aim the missile?
Just a thought.
If you can work out the flaws in the Gyrojet series of weapons, yes, this is absolutely possible.
With the real-life Gyrojet project, production was rushed, machining flaws made the ammunition very inaccurate a lot of the time, the gun reloaded slowly (in the case of the pistol version) due to lacking a real magazine, and even the most-functional rounds weren't as accurate as pistols at the time, although individual rounds hit with about twice the energy of .45 ACP.
Solve the production flaws, give them proper magazines, and maybe make them with modern-day technology, and Gyrojets are your best friend here. The low accuracy likely doesn't matter as much onboard spaceships, where engagement ranges are limited.
Given that you can make rockets at this size, as well as rockets the size of a Javelin anti-tank missile (as opposed to a Javelin antiair missile - an entirely different beast), it's reasonable to believe you can make one that's in between the two size-wise.
In principle one can use an atlatl
A spear-thrower, spear-throwing lever or atlatl (pronounced /ˈætlætəl/ or /ˈɑːtlɑːtəl/; Nahuatl ahtlatl [ˈaʔt͡ɬat͡ɬ]) is a tool that uses leverage to achieve greater velocity in dart or javelin-throwing, and includes a bearing surface which allows the user to store energy during the throw.
It may consist of a shaft with a cup or a spur at the end that supports and propels the butt of the spear. It's usually about as long as the user's arm or forearm. The user holds the spear-thrower in one hand, gripping near the end farthest from the cup. The user puts the butt end of the spear, or dart, in the cup, or grabs the spur with the end of the spear. The spear is much longer than the thrower. The user holds the spear parallel to the spear-thrower and going in the other direction. The user can hold the spear, with the index and thumb, with the same hand as the thrower, with the other fingers. The user reaches back with the spear pointed at the target. Then they make an overhand throwing motion with the thrower while letting go of the spear with the fingers.
A spear-thrower is a long-range weapon and can readily impart to a projectile speeds of over 150 km/h (93 mph).
It is doable to adjust it to fit a missile instead of a spear.
Self-contained rocked
Electronics are cheap and lightweight, machinery and fuel mean mass (physical burden) so we try hard to use the mass budget effectively.
The rocket has a small display where you select the target from whatever the rocket "sees" at the moment.
Then you throw it in the general direction of the target (you may as well throw it wherever you like, it will take the proper turn at the expense of some more fuel).
The rocket engine ignites when feels safely away from friends. If it doesn't get to a distance safe to ignite, it doesn't ignite so it can be picked up and retried.
You are free to throw it however you see fit the purpose and the possibilities - as a knife, as a spear, backwards (holding the nose)...
Or just put it with engine down on the ground and run away. The rocket (when engaged) will start as soon as YOU are at a safe distance.
Throwing saves you the launcher tube - it is heavy and is either disposable (heavy * the number of rockets + leaves artifacts or the need to carry them back) or multiple use (requires an additional effort to reload and an additional effort to select the target as it covers most of the rocket).
It's pretty much impossible.
To safely launch a missile you must be able to get it far enough from you that when the main motor ignites it doesn't fry the guy who launched it. Let's look at that Stinger missile mentioned in another answer--5 kilograms for the missile. The main motor ignites 9 meters from the guy who pulled the trigger. Can you lob a 5 kilo weight that far? (Remember, this is at a substantial angle upwards, not 9 meters along the ground.) Given human muscle performance you need to use a booster of some kind to throw it far enough away to be safe. In practice the lightest way to accomplish this is a fast-burning solid fuel rocket motor--hence almost all such weapons use this approach. (There are a few that throw something else out the back--special-purpose weapons that can be fired out a window without killing the operator. You pay a weight penalty for such rounds.)
There are hand-launched systems, but they are propeller-driven drones, not missiles. Propellers do not have a safety distance, you simply need to get them up to minimum airspeed, they can safely fly right out of your hand.
I think a lot of answers here are assuming an anti-tank or anti-air weapon when they say that without the tube, backblast is a big problem. I generally agree with them.
But if you're talking anti-personnel - especially anti-personnel in a zero-gee / low-gee environment - I could totally see this working.
For an anti-personnel weapon, you can get away with much smaller missiles. They aren't trying to use kinetic energy to punch through several feet of high tech armor. These "missiles" wouldn't be much larger than standard rifle rounds are now. They're doing the same job, with roughly the same tools.
For a low gee environment, a hand toss can give you the ability to "fire" your missiles silently, move for several seconds, and then have the rocket motors kick off. You could do one-person flanking movements where you throw a handfull of missiles around a corner, and then run to the next corner to engage the enemy just about the time they realize they're being attacked from the first direction.
So figure out what the use case is, and then you can figure out what the best tool is for the job. I think anti-personnel, low-gee combat could be a workable use case for this tool.
I'm not totally sure about what setting your world is in, but I can see a lot of room for error with hand-launched missiles. It was touched upon, but nobody mentioned the exhaust of the missile blasting right into the soldier's (or otherwise missile-throwing person's) face and chest. In our own world, some anti-tank weapons and rocket-propelled grenades could give someone severe burns if they are fired in a confined space, such as if the breach is too close to the wall behind the person firing the weapon. In an open space, standing directly behind the breach poses the same danger.
Maybe the technology of your world makes it so that this problem with the exhaust is circumvented, but know that missiles generate an immense amount of thrust. The person throwing the missile could, perhaps, run clear of the missile's back-end, but that is still a risky endeavor. Another solution may be to have the missile ignite its thrusters after a traveling a certain distance or after a designated amount of time has passed, but I'm not confident with the idea that a missile could be easily thrown in such a way that it reaches a safe distance; and it's still risky.
