TL:DR: Yes, it depends on the gravity of the planet.
Water pressure exerted on an object is dependent on the weight of a column of water above the object.
Weight is a force dependent on 2 things, mass and acceleration due to gravity (f = ma).
The mass of that column is its volume multiplied by the density, and as (liquid) water is incompressible, the density is 1kg/L. On earth, the acceleration due to gravity is ~10m/s²
This means if you were 1m under water and had an area of 1m² there would be 1m³ column of water pressing down on you with a mass of 1,000kg. Multiply that by acceleration due to gravity, that’s 10,000N of force, divide that then by the surface area (pressure is force/area) you have your water pressure. The pressure will increase with depth as more water will be above you pressing down.
As I mentioned, water is incompressible, meaning its density won’t change. The only other variables are depth and acceleration due to gravity.
Another planet may have higher or lower gravity dependent on that planets mass, meaning the pressure per unit depth will be higher on more massive planets and lower on less massive planets than Earth.
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