Yes you can safely drink the tap water on cruise ships as it's just as clean or maybe even cleaner than the tap water at your own home.
The tap water on cruise ships goes through rigorous filtration and testing, all of which are overseen by U.S. and European heath agencies.
So the tap water on cruise ships is safe for human consumption so there's no need to bring bottled water on the cruise ship.
The cruise ship tap water is likely better for you and your health than bottled water ever will be.
Cruise ship tap water usually comes from tanks that are filled before the cruise ship sets out on the ocean and it's basically city water.
Cruise ships get their fresh water through other ways such as evaporators, reverse osmosis and shore-side water supply.