Did I damage my belly button? Is it permanent?

You may have.  You should not tan ever! The rays of the sun can darken the scar. On larger scale long term tanning can lead to skin cancer several years later

One tanning session will have no effect on the long-term outcome of your scar. Further UV exposure should be avoided.

Thanks for you question but pictures are needed. You shouldn't be taking the sun 6 weeks post-op. I don't know if your palstic surgeon gave you post-op instructions. Maybe it's just a sunburn but you should call your plastic sugeon and ask what to do in this case.

Thanks for your question.  Tanning a pink scar should be avoided.  A pink scar if tanned can stay dark permanently.  All you can do at this point is to avoid additional exposure.  

Without photos, it's hard to even begin to answer your questions.  In general, skin that has been stretched acutely will tan/burn more easily.  That's why we recommend tummy tucks, breast augs, facelifts, etc be careful not to get too much sun exposure in the first 6-12 months after surgery.  If you have hyperpigmentation, it might not end up being permanent.  Show it to your PS.  He or she may put you on a bleaching agent called hydroquinone.