As Nik says, you can change the prop when afloat but it is not an easy job - it would certainly be better to have a dry run
before attempting it on the water.
Once you know how all the bits go together though, it can be useful to be able to do it on the water - one day you might have to! Bear in mind that you will have to perform the task somewhat blind, as the prop is almost certainly going to be under water! However, if you have got one or two hefty mates that you can put up in the bow, you might get it above water level. Note also that an extended swim platform can make it very difficult to reach the prop. The trick to not losing any of the bits (you will drop them!) is to put a large mesh bag around the outdrive and always carry a spare everything.
Lastly, you can get plastic floating prop wrenches too but don't bother... it won't help you if you drop any of the other bits and, if the retaining nut is very tight, it will just strip the soft thread on the wrench. Carry a metal wrench and a mesh bag!
Oh, and when you say 'how would I know if it's spinning the prop' I take it that you mean whether the prop is slipping on the hub? This tends to be an all or nothing problem - if the prop hits a submerged object, the hub is supposed to shatter to save the drive (but not the prop itself). If this happens you will need your spare prop and hub and your new-found ability to change the prop on the water! If you are concerned though, you can inspect the hub for damage / wear when you remove the prop. However, note that there will always be a bit of prop slip through the water itself as the prop is not 100% efficient. This is perfectly normal and, for a sportsboat, prop slip will probably be in the order of 85%-90%. You can determine the amount of prop slip by calculation but you must first accurately establish the boat speed and revs at WOT, check what pitch prop you are running and find what reduction ratio the sterndrive is (it will be stamped on the gear case). From this data you can calculate both the theoretical and actual forward motion of the boat per revolution of the propeller - the ratio of the two is the prop slip.