It would be like trying to run down a Jack Russell Terrier with a train.
Sure, you could do it, but the Jack has to not be paying attention for it to happen.
If the propulsion plant of the destroyer is off-line, how long would it take to spin up the engines to get underway? Is there any condition where the propulsion plant would be off-line while the ship is not lashed to a pier?
This destroyer has two screws (propellers), meaning there are two main engine rooms. (Looking it up, each shaft has two gas-turbine engines.) There are multiple backups, as you would expect, redundancy is built into the design.
Assuming you didn't suffer some kind of engineering casualty to both engines, it's possible you might take one shaft offline for maintenance. You would never do it in a situation where you might endanger the ship by not having her fully operational. This might be in a hostile situation, or in a navigational situation such as a transit through a channel. Navigating a busy sea lane off the coast of Japan might also be considered "dangerous" from this standpoint.
I would think that any planned maintenance would've had to be authorized by the Captain, prior to going to sleep. This would be in his nighttime standing orders. Any un-planned maintenance that was required to repair a power plant would've resulted in someone waking the Captain to inform him of the situation.