Not only that, but if the Earth could somehow retain a disc shape, gravity on such an Earth would behave quite differently than what we experience. As we moved south (which, on the flat-Earth model is toward to outside of the disc), we would encounter more and more force drawing us back toward the center. IN the Southern Hemisphere, or outer circle, we would increasingly experience the feeling of going uphill when, in fact, we were not.
So how do the flat-Earthers deal with the problems posed by gravity? Simple: they deny that gravity exists at all.
In the flat-Earth world, the effects that we see from gravity are easily explained by buoyancy and density. Things that are more dense move downward past things that are less dense, and things that are less dense move upward through things that are more dense.
Thus, hammers fall, oceans stay put, and helium balloons float. Really simple. And completely wrong.
Now, I'm not a physicist. I can't give you all the details of how gravity works. For that, I suggest you read a good book on elementary physics, or if you're really ambitious, Newton's Principia. But there are some things so obviously flawed in the flat-Earth denial of this most basic of physical laws that pretty much any thoughtful layman can easily see right through it.
Let's start with this: the existence of gravity is an established scientific fact. It isn't "just a theory." There are theories of gravity, which attempt to describe what gravity is and why gravity is, but that gravity is is not in doubt. That's why there is the Law Of Universal Gravitation.
And we need to get another thing straight in our minds: gravity is not a one-way street. It isn't just the Earth pulling things toward its center; those things that are being pulled are doing a little pulling themselves.
So, let's kill the density argument very quickly. Take two steel balls, one an inch in diameter, the other three inches in diameter. They are of equal density, but the three-inch ball has 28 times the mass of the one-inch ball. drop the two balls into a body of water (I don't recommend a tub unless you want to damage it). Which one makes a bigger splash?
Same density, different mass, different amount of force applied by (you guessed it) gravity.
So now let's work on buoyancy. Buoyancy is what makes a rubber raft float on a lake, and a helium balloon float in the air. But the helium balloon is not defying gravity any more than the rubber raft is. The Earth is still pulling on the balloon, and the balloon is still pulling itself toward the center of the Earth. But it's being pushed as well, the exact same way the rubber raft is being pushed to the surface by the density of the water.
But at some point, both the raft and the balloon will stop rising. The raft meets the point where it is not longer being pushed up by the water. The balloon either reaches the point where the air is no longer more dense than the gas inside it, or it bursts.
Both are the result of gravity. It's gravity that makes air pressure higher close to the ground than several thousand feet up. Buoyancy means nothing without gravity.
So let's eliminate buoyancy altogether, and see how gravity works in a vacuum. Here's a video from the BBC:
Not only do the feather and the bowling ball fall at the same rate, but notice how much damage the bowling ball causes, while the feather causes none at all. Density, buoyancy have nothing to do with it. It's all about that mass.
Flat-Earthers claim that no experiment has ever shown that two objects attract each other except by magnetism. They discount the Cavendish Experiment as a failure, or a fraud, and claim that it has never been duplicated. But even physics students can prove the effects of gravity.
And so can you, using your phone. Your phone has a gravity sensing system that uses accelerometers. If you try to attribute the phone's ability to detect rotation to acceleration from the turning of the phone, however, you're left to explain how the phone knows its initial position when you first turn it on (even from the state of having removed all its power). It detects this state from the acceleration of Earth's gravity.
We haven't just proved gravity, we measure it and use it every day. And because gravity exists, the flat Earth doesn't.