Child support has nothing to do with it. If you have a legal right to have the child with you, you must have some sort of document ( a custody order, a parenting plan etc.) If you are simply on the birth certificate, you are not yet the legal father if you weren't married to the mother until the court issues an order.
It looks like you can't. I suppose they might allow a notarized statement from the other parent allowing you to go on the cruise. If that is not feasible, then I think your only option is to go to court and get an actual custody order.
Child support and time-sharing are two different things and are not dependent on each other. If there was a divorce or a paternity action, you should refer to your parenting plan provisions/final judgment concerning the same. It is not clear from your post if you are a father or a mother of the child and whether you were ever married to the other parent. Generally, if the parents were not married and no paternity action has been filed, then the mother has all the time-sharing and decision-making authority over the child and the father only has a legal obligation to financial support the child until a paternity action has been commenced.
Get a notarized letter from the other parent giving you permission to take he child out of the country. This is what is required to fly, take a cruise or anything else whether you are married or not.
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