I want to go swimming but I don't know if its okay to go swimming when my belly ring hasn't healed fully..

Not really a good idea unless you are wishing for some kind of swell infection. Don't do it as you have no idea what is lurking in the water that can infect you.

Swimming pools are full of bacteria and chemicals so itd be safer to not swim in one but you can just pop a waterproof plaster on and then clean it with your cleaning solution as soon as you get out. As for the sea, it depends, if its really high in salt then id say no because it can dry it out and the sea is full of nasty things but it can also help healing, I went on holiday after getting mine done and it was absolutely fine going in the sea, just make sure you wash it afterwards as you would normally. Hope this helped.