"Fueled By Water" is somewhat misleading. I assume you're talking about Hydrogen Fuel Cell vehicles, which actually run on hydrogen (though, ideally, that hydrogen would be stored in the form of water in the tank on the car).
In answer to your question, no, more water on the roadways isn't going to make it rain all the time. Water vapor is more significant than greenhouse gases, though, in terms of global warming.
Our cars currently produce water vapor as a side effect of combustion engines, in addition to other things...and currently, the amount of water vapor contributed by all forms of combustion, all over the earth, is about a 10,000th of the evaporation occurring naturally, so it isn't going to make a significant difference...first because the vapor is already being produced with combustion engines, and second because even if we weren't, cars and combustion just don't produce much vapor compared to what the earth does naturally.
The technology won't be marketable for years yet, and most likely, we won't be using water as the fuel source, we'll use compressed hydrogen so we don't have to separate out the H2 from the 02 in the engine...that means entire infrastructure changes before we can actually see the change (because if cars are going to run on hydrogen, you're going to have to build hydrogen stations to replace gas stations, or at least as a part of them as people transition over...no one will buy a car with nowhere to fill it up).
Of course, the major current problem with cars running on hydrogen is that no car can, currently, run directly from water...they all have to run on liquid or compressed hydrogen...and we make that hydrogen by burning coal, which isn't very green anyway.
I've driven a fuel cell car, though, and they are nice. Felt similar to a gas car in ride, not nearly as smooth as all-electrics.
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Cars fueled by H2O
I walk dogs everyday - I don't wanna smell like wet dog ALL THE TIME! lol 
