You might have to contact the cruise lines individually, and multiple times (sometimes different CSRs may give different answers).
My gut says no. You would have to check the bike like on an airline. The bike would have to be placed in the hold by the porters then brought to your cabin. At each port you would have to pass through a metal detector with the bike.
You might be better off investigating local bike rentals at your desired ports of embarkation. Also realize that some islands do not have the best of roads and some "exciting" drivers. And, hey, maybe you would enjoy a different type of adventure for a while.
All I ask is a tall ship and a star to steer her by; John Masefield
Carnival: Glory 2004, Destiny 2008, Splendor 2009, Freedom 2011, Valor 2012, Dream 2013
Celebrity: Summit 2011
Princess: Ruby 2010, 2014, Caribbean 2013, Coral 2014, Regal 2014
Star Clippers: Royal Clipper 2015
"Safety Concerns: Due to "Safety Concerns", the following items cannot be utilized on board nor are permitted on board: Bicycles
Scooters (unless medically necessary)
Scuba Tanks (empty or full)"
Yes, but only if you buy the kind of bikes that many RVers own and use. They are called collapsible bikes or folding bikes. They fold up completely flat and then you carry them in a bike carrying bag or backpack. Their are several makes and models to choose from. The bikes tend to be a bit on the expensive side, but I have seen some recently at about $125 to $150 each. I have brought them on nearly all trips and walk on and off the ship with the bike in a backpack; once on shore, fold it out and go for a nice ride. Of course, you can never use a bike onboard the ship.