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.