A nuclear carrier might use a system like this to generate fuel for its planes or accompanying vessels.
Apart from that, perhaps a diesel-powered ship might generate JP5 for the helicopters and drones it operates?

It would be far more practical to just use nuclear power for the ships although as a method for producing fuel for aircraft it has merit.

Is waste heat from a gas turbine hot enough for this process?

At $1.58 / litre I do this to make fuel for my car which costs about that much. I could just drop a couple of hoses into the canal outside my house and fuel up. Assuming I had a batmobile, which lets face it, we all want.

Have a conventionally fuelled resupply ship fitted with a very large scale version of this equipment. Each ship could potentially serve multiple squadrons, and with no combat duties they can concentrate on fuel production for the majority of time at sea. A nuclear powered craft could probably handle the power requirements, especially considering these ships don't need to be particularly fast.

So... if the Navy now see merit in this, when will we start using it for cars?

I am glad they added that last paragraph because this is very much the dumbest idea and article I have ever read.
Why don't we just research running our ships on "happy thoughts" because that is more likely then a ship"refueling itself using sea water as fuel".
Seriously I do understand all the process they described, the whole concept is beyond absurd.

Where does the energy come from? How about off shore wind turbines.

The obvious and only way this makes sense is that nuclear carriers would produce fuel for their planes this way.

Other people at the same lab have been doing Cold Fusion work since 1990 so that's where the energy will come from. It will be D2/ Pd or H2 Ni gas phase low energy nuclear reactions producing both heat and electricity to power the reaction. These cold fusion reactions run slow and steady and don't like variable loads. Powering a steady chemical conversion system that then powers the variable loads: turbines; small boats and drones makes perfect sense.