OK, so I did as was suggested and tried checking the tire pressure with the tire rolled so that its valve stem was high, rather than low.
Same result--a small amount of green liquid came out, and couldn't get a pressure reading. I then tried using a hand tire pump (not a compressor; just a pump for pumping up bicycle tires) to put air into the tire. No go. Couldn't force any air into the tire.
The green liquid is thin in consistency, and somewhat slippery/oily.