Sure they do - at least if you follow colloquialisms. The MLWS displays its output on the IP-1310, which is commonly known as ... the RWR display.

And it WILL show you any missile that it can detect within its limits, be it passive or active.

I don't see how it can detect a passive (as in not emitting radiation) missile. I'm neither a real pilot or an avionics engineer so if it can please tell me how.

I think it detects the IR signature of a missile launch. That's why the A-10 RWR goes haywire with Maverick and Hellfire launches.

Some combination of IR+UV detection of the rocket plume.

Well, there are three types of MWS. Radar, IR, and UV. The system installed on the A-10C (and every other MWS eqipped jet I know of) is UV.

The sensors in UV systems are nothing more complicated than CCD/CMOS sensors tuned and filtered to pick up light in the UV part of the spectrum. They detect missile launches via the UV radiation emitted by the rocket motor, which is why they can only detect a missile if one of the sensors is looking in the right direction at the time of launch, as soon as the motor has burnt out the system can't detect the incoming missile. Now as all the system does is pick up on sources of UV radiation, any UV source could (in theory) cause the system to give a launch indication. Good software with a database containing the UV signatures of known sources can help recduce false indications, but nothing can ever stop them, there are just too many variables.

Between IR and UV systems, a UV system is least prone to producing false indications, especially at low level. IR systems also require cooling (just like an IR AAM) and so have a larger footprint on the aircraft, require more maintenence and are more expensive. IR systems however are more effective at high altitude and can (again in theory, and only at high altitude) detect an incoming missile after it's motor has burnt out using the heat generated on the missile body from friction of the air. As you can imagine, at low level an IR system is very prone to false indications which is, along with their higher cost, the main reason why UV systems tend to be preferred.

Radar systems, are least prone to false indications and can obviously detect a missile if its motor is burning or not, and even (in theory) determine if it's actually directed at your aircraft. But they are much more costly, larger, heavier and complication than their UV and IR bretheren.