Did I damage my eyes in a tanning booth if I forgot the goggles?

This is impossible to answer because we don't know the strength of the light. But it is unlikely to have harmed you significantly. Ultraviolet light can harm the cornea, increase the chance of cataracts, and probably increase the risk of macular degeneration in the retina, among other problems. These injuries are unlikely to occur because of a 15-second exposure but rather a lifetime of exposure. Most likely, the dose you received was quite small compared to what is required for injuries to these tissues occur. Beware of tanning salons and exposure to UV light, though, as these can also increase your chance of several skin cancers.