No, you can't. Two practical reasons and one legal.
As a brand new diver, you should avoid trying too many things at once. Task loading can easily lead to confusion and mistakes on the life-support tasks. So don't try more than looking at the scenery for your first dives.
Second, if you were to manage to catch something, would you have a way to reasonably keep it alive for the remainder of your trip, and transport it home? Keep in mind that in a tropical climate, the air temperature is likely warm enough to cook a fish in a small tank unless it is kept in an air-conditioned room.
Finally, the legal aspect. While some creatures may be legal to collect, many are not or require special permits. And to import any of these back into the U.S. requires more permits, as well as arrangements to meet a representative of the Fish & Wildlife service at your port of entry. While all of this is possible to arrange, it's more than a hobbyist can realistically arrange. It's even a pain for the professionals who do it regularly.
I've heard of people ignoring the laws and doing this anyway. The penalties are pretty drastic if you get caught, and with current airline policies, you're pretty likely to get caught.
-Mark