A company called teh Wedding Experieience will organise all weddings for RCI - www.theweddingexperience.com - you get married on board in certain ports, and in others you go to the local town hall. Forget all romantic notions of the captain marrying you as this does not happen.
Although RCL may not have the captain wed you, the captain DOES wed you on Princess - they have been doing this since 1998...
If you want to know more, check out http://www.princess.com/news/article.jsp?newsArticleId=na595&submit=p.
Captain marries you during a P & O Cruise. (Florist & photographer on board). P & O are great.
I've just been on the Mariner of the Seas and there was quite a few people getting married so I would suggest you talk to Royal Caribbean office.