I wonder if a rocket that uses electricity from internal batteries to heat water in its tanks to high pressure would be able to use that pressure to take off. Is it possible?
I guess technically it could get off the ground, but it wouldn't be able to get into space. The energy density of rocket fuel is massively larger than batteries. Moreover, rocket fuel serves both as fuel and reaction mass. A fundamental difference between rockets and other uses of batteries, such as cars, is that rockets have to carry their own reaction mass, while cars use the ground as their reaction mass. This means that they have to be very efficient in their use of reaction mass. And while a car being inefficient can be solved by simply carrying more fuel, piling more fuel onto a rocket just makes it heavier and makes it need more fuel and reaction mass. Past a certain level of inefficiency, you get into a fatal loop of more fuel needing more fuel which needs more fuel, and you can't get to space no matter how much more you pile on.
I believe this method is much better than building a rocket with all the pressure inside it before launch.
Why? When we pressurize fuel before putting in the rocket, we use Earth-bound energy sources, which means that we don't have to take those energy sources with us. If we're adding pressure midflight, we have to take the fuel to do so with us, and we need fuel to lift that fuel off the ground, and we get back to the issue I mentioned before.
Using electricity from batteries also seems a better idea than using any other type of fuel for heating.
When I think about this type of rocket, powered by water and electricity, I imagine it's just a matter of engineering.
Not really. It's also a matter of physics. A battery works by having chemical reactions produce ions that produce electricity. It makes more sense to have chemical reactions directly producing energy, and rocket fuel reacting with oxygen is one of the most energetic chemical reactions there is.
It would be as difficult to build it as any other rocket, yet it looks much cheaper and more environmentally friendly.
Batteries are much more expensive than gasoline. The reason batteries are more environmentally friendly is that the chemical reactions that power them are much more reversible than the ones involved in burning hydrocarbons, and so while hydrocarbons are basically single-use, batteries can be recharged. If we had a good way of turning carbon dioxide and water into gasoline, gasoline engines would be more environmentally friendly than electric cars. For the most part, a rocket is single-use no matter how it's powered, so you lose the major upside.