I want to create to create a ship that does NOT rely on rotational gravity to have a constant downward force on everything. The people built a g-force generator that does NOT induce gravity, just a force.

So, can a gravity generator that uses energy to accelerate objects towards the generator work?

If there is a 100 joules of energy that supplied to the generator, it could accelerate a 1 kg object down under 1 $m/s^2$ for 200 seconds. This situation does not regard efficiency or the mass of the generator itself for simplicity. The mathematics are based on energy, mass, and velocity ($E = 1/2 cdot M cdot V^2$).

Remember, this would be in the void of space. Would this be possible under real physics or would I have to implement a fifth force for this to happen? If it can happen, could it have an inverse square relationship with distance like real gravity?

It can work in any way possible if there is one, but the generator has to work regardless of acceleration of the generator itself. For example, the generator works whether it has a total acceleration of 0 $m/s^2$ or 100,000 $m/s^2$. It can work combined with other generators or actual gravitational pulls.

You could simulate gravity using electrostatic attraction.

I cannot run the numbers you provided, but electrostatic attraction can be strong in proportion to the charges involved. It is the same principle that causes cat hair to be stuck to a rubbed balloon. It would work even better in space because space is dry and charge could not leak off into the humid air. In an atmosphere, charge can equilibrate across a space full of gas by breaking the gas down into glowing conductive plasma - this is a spark. In space there will be no equilibrating unless the charged surfaces come into contact with each other, and you can prevent that by coating your spacefarers in nonconductive full body skin tight latex suits.

Using charge for attraction in space is not purely zany. NASA has a scheme for an "E-glider" - a spacecraft which makes use of these electrostatic forces around asteroids to maneuver. Unlike gravity which only pulls stuff together, charge is more versatile and can be greater or lesser, positive or negative. In the example the charged vehicle maneuvers relative to the asteroid using manipulation of charge.

https://www.nasa.gov/feature/e-glider-active-electrostatic-flight-for-airless-body-exploration

There was no mention of the skintight latex suits in this NASA article, but I feel they were strongly implied. The spaceships will need latex suits too.

Those already exist, they are called rockets.

When astronauts take off from Earth they are iften faced with high G forces (up to 8g if I recall correctly). That is not coming from rotation, nor from the Earth.

In fact, if you wish to go anywhere far from here, a constant rocket acceleration might provide you with some comfort. A constant 1g from the engine will keep your bo es healthy and take you to other star systems relatively quickly.

There's a reason why the centripetal gravity ring is so popular

Here's your spaceship:

Note: I am not an artist.

You start by accelerating weights

If you accelerate a 1 $kg$ weight downwards at 1 $ms^{-2}$, you'll exert a force of 1 $N$ on it. Now, that 1 $N$ gets distributed across the mass of the entire spaceship. Let's say it weighs 50,000 $kg$. This means that the acceleration will be 0.00002 $ms^{-2}$, roughly 2 millionths of Earth's gravity. You won't even notice that.

In order to provide Earth gravity to a spaceship that big (which is a realistic size for what you're looking for), you'll need 500,000 $N$ – the equivalent of launching a 500,000 $kg$ mass down at 1 $ms^{-2}$ (which is rather impractical, since it weighs more than your ship) or launching a 100 $kg$ mass down at 5000 $m s^{-2}$.

Now you're accelerating your whole ship

You're now launching a huge mass at a huge speed downwards. As Renan mentioned in his answer, this is starting to look like a rocket engine. But the problem remains: now that you've sent that thing flying downwards, you're not getting it back. You've just lost 100 $kg$ and your gravity has ended as quickly as it began.

So you need to continuously launch huge masses at huge speeds. This is going to have the side effect of pushing you in the other direction, really fast. In fact, to get 10 $ms^{-2}$ of pseudo-gravitational acceleration, you need to accelerate at 10 $ms^{-2}$ upwards.

So unless you want your gravity generator to fling the ship around (...it's literally a rocket engine!), you'll have to flip the ship around and go the other way for a while. That way, you spend a few minutes accelerating one way, then have a short zero-g break while you rotate, and then spend a few minutes accelerating the other way. Don't try this while docking.

And now you're swinging back and forth

So you ideally want to avoid the zero-g period while you flip the ship. So you keep the gravity generator (aka rocket engine) on the bottom of the ship and leave it running while you turn. So you spin around in a spiral pattern, always accelerating upwards relative to the people.

You want to make the spiral as circular as possible, because a perfect circle wouldn't change the trajectory of the ship, whereas anything else would. So you add side thrusters and spin around in a controlled circle, spending equal time accelerating in every direction.

And now you're orbiting a point.

Your main issue now is that you're throwing out all this mass and not getting anywhere. Bear in mind that you have to carry all these weights with you everywhere you go just to have gravity.

So instead of constantly launching the weights, you keep one big weight, heavier than the ship (maybe it's a fuel tank or something), and tie the spaceship to it using a long cable. That way, the spaceship swings around the weight like a sling, creating the same circular effect, but you don't burn any fuel.

But your ship pulls on the weight in the middle, and swings it from side to side. So for balance, you put another equally heavy ship on an equally long cable on the other side. Now it's like a 2-bladed ceiling fan: the symmetry prevents it from wobbling.

Now, you probably see my point. If you don't want your generator to fling you chaotically around space, the best solution is a rotating centripetal gravity system. I know that's not what you asked for, but your gravity generator is literally a rocket engine. If you want gravity while the ship isn't burning its engines, you need one of these: