As stated in the The Internal Pressures of the Planets, if we use the surface gravity of a planet as an approximation to its internal gravity, we can show that the internal pressure is proportional to the square of the surface gravity. But, as just described, the internal gravity varies from place to place, dropping to zero at the center. So how can we be sure that the original approximation is in any way near the correct value?
      Let's take a more accurate look at how the calculations would be done, if we were willing to use the correct kind of mathematics, integral calculus, to do them. (Please note that you do not need to be familiar with calculus to follow this discussion, nor are you expected to be able to reproduce this on your own. This is just to show you how things are done, and how the results compare to the simpler approximations that were used the earlier discussion.)
      Let's imagine dividing a planet into a nearly infinite number of very thin layers, each with a thickness, dr (equal to the radius of the planet divided by that nearly infinite number of layers), which is so thin that it is negligible in comparison to the size of the planet, and the density of the material within that layer can be thought of as being essentially constant. (Note: In calculus, writing a quantity as "d-something" specifically means that we are using only a very thin layer of "something".)
      At the top of the layer, the weight per unit of area which is compressing the material, and the pressure opposing that weight, would be given by w_r and p_r, the subscript r indicating that those are the values at that particular distance from the center of the planet, r. At the bottom of the layer, however, the two values would have to be slightly greater, because there would be a certain extra weight within the layer, dw, which would be equal to the density of the material in the layer, d_r, multiplied by the volume of the layer, which would be dr times the area of the layer. Since we are defining everything in terms of the weight or pressure per unit of area, the area of the layer can be taken as one unit, so we can ignore that. In that case, the extra weight, dw, would be equal to d_r times the local value of the planet's internal gravity, g_r, times the thickness of the layer, dr. Now, you might note that the letter "d" is used here in two ways -- as the density, and as an indicator that we are using a small amount of some unit. For that reason, we will replace that letter, in talking about density, with the Greek letter rho (r), as is often done in such calculations, so that

dw = dp = r_r g_r dr,

where we have indicated that the increase in p_r, which is called dp, will of course be equal to the increase in w_r.
      With these definitions and statements in mind, we can now calculate the pressure and weight at various depths inside a planet simply by adding up all the values of dw and dp, starting at the surface of the planet, and going down into the middle. This could be expressed as

w_r = p_r = w_surface (= 0) + dw_{first layer down} + dw_{second layer down} + and on and on and on,

adding up however many layers we need, in order to reach radius r. Obviously, if the number of layers is very large (and, since we are specifiying that they are very thin layers, so to go down any substantial distance would require a very large number of layers), this would get very messy, so we indicate the sum of all the terms involved by a special symbol, the integration symbol, , which means "add up all the individual values, starting at the surface, R, and going down to radius r", and write the above equation as

w_r = p_r = dw = r_r g_r dr.

      Of course, writing this formula in a simpler way doesn't mean that it is all that easy to actually do the arithmetic, as we have to know, in order to do this calculation, just what the values of r_r and g_r happen to be at every point inside the planet, and that is not necessarily something that we can know with any certainty. But to give an idea of how things would work, let's suppose that the density, r_r, happens to be the same everywhere (in other words, we have a "uniform" planet), which would make it equal to the mass of the planet, divided by its volume

r_{uniform planet} = M / (4/3 p R³) = 3 M / 4 p R³.

Under those circumstances, the gravity at a given place would be, as already discussed, directly proportional to the gravity at the surface of the planet, or

g_r = g_R r / R,

so r_r g_r dr

would become (3 M / 4 p R³) (g_R r / R) dr.

But g_R is equal to G M / R², so this becomes

(3 M / 4 p R³) (r G M / R³) dr,

or, rearranging like terms,

w_r = p_r = (3 G / 4 p) (M / R³)² r dr = (3 G / 4 p) (M/ R³)² r dr,

where the constant terms have been removed from the integral sign, because they don't affect the integration.
      Up to this point, although the mathematics appears difficult, all we have been doing is defining things, and rearranging them, and no actual arithmetic has been done. At this point, however, we need to do the integration, if we want to get any particular result, so if you are not familiar with integration, you will have to take my word for it that the term inside the integral sign becomes

1/2 (R² - r²) = 1/2 R² (1 - (r/R)²).

Inserting this in the previous equation, in place of the integration, yields

w_r = p_r = (3 G / 4 p) (M / R³)² 1/2 R² (1 - (r/R)²),

or, rearranging and combining terms,

w_r = p_r = (3 G / 8 p) (M / R²)² (1 - (r/R)²),

or, since the surface gravity, g_R, is equal to G M / R²,

w_r = p_r = (3 / 8 p G) g_R² (1 - (r/R)²).

      In other words, at least in the case of planets with uniform density, the pressure at various points inside the planets are directly proportional to the square of the surface gravity, exactly the same result that we obtained when presuming that the gravity inside was everywhere the same as at the surface. This does not, of course, mean that things will work out so nicely when the density changes from place to place, and the way in which the pressure changes as you go into the planet would not be the same, if calculated using the incorrect assumption of a uniform gravity, as it is when calculated correctly, but it does show that if we were to try to compare the pressures inside various planets, squaring their surface gravities isn't a bad first approximation.
      Finally, by having calculated the actual value for pressure at various points inside a uniform planet, we can calculate how the pressure changes from place to place, as shown in the diagram below. Near the surface, where the gravity is close to the surface gravity, pressure increases more or less uniformly, but as you near the center of the planet, where the gravity approaches zero, the pressure levels off, and becomes nearly constant in the central parts of the planet.

The pressure inside a uniform planet.
The numbers along the bottom show the fractional distance from the center of the planet, r/R.
Pressure, increasing upward, has a maximum value of (3 / 8 p G) g_R².