Do rainbows show spectral lines from water?

The water droplets that create a rainbow are not emitting the light that you see in a rainbow; if they were, you would see a glowing cloud of consistent color, not a rainbow. The rainbow is formed by sunlight refracting and reflecting through water droplets in the air; the water refracts through the "front" of the drop, reflects off the "back," and refracts again on the way back out. The refractions are what separate the colors, since different wavelengths of light refract to different degrees. If you used devices capable of imaging in other wavelengths of light, you'd see further bands of "color" beyond the red and violet sides of the rainbow, resulting from the infrared/ultraviolet (and other wavelengths beyond those) radiation in the sunlight.

So in short, the full-spectrum appearance of the rainbow is due to the fact that the source of the light (the sun) is a thermal blackbody and emits a blackbody spectrum.

No.

As has been said, the raindrop is not emitting the light, it is just acting as an optical device that deflects light emitted by the sun. However, the spectral lines you would expect to see in sunlight refracted by a prism will not, repeat NOT, be seen. The mechanism that produces rainbows is very different than the mechanism that produces a spectrum with a prism.

I've written a much more detailed description here: What makes a rainbow happen?. A summary is that each raindrop deflects light at all angles within a cone that has a half-width of about 40° (for spectral lines near violet) to about 42° (red). Each cone is brightest, by far, at the very edge of this cone, which is why we see colored bands. But each band we see is actually a composite of all colors from the one we perceive, to the start of ROYGBIV.