If the boat was run any significant amount, the alternator would have topped off the new battery. That being said, hopefully you didnt buy a deep cycle (marine) battery? Those are intended for long, slow draws (trolling motors and stereo systems) and not short, high amp draws like starting a V8 engine. If thats what you have, then it may be a big part of your problem.
I cant tell you how many times I have seen this happen on my boats or others. Dozens. Have replaced 2 starters, so they do go out and cause these symptoms... but they are the true cause <10% of the time. So I agree with those above who are telling you that replacing the starter at this point is premature.