It's hard to have a definite answer because of the unclear definition of "a square inch of the world". Nevertheless, the exact number is going to be a really really big one.

To talk about the number of atoms, it is necessary to introduce the Avogadro's number, which is defined as the number of units (atoms, molecules, etc..) in one mole of any substance.

Here the Avogadro's number is measured to be around 6.02 times 10²³, and mole is related to Avogadro's number and the mass or density of the units.

To give an example, one mole of water is referred as one Avogadro number of water molecules, or 18.015 grams of water (the atomic weight of a water molecule is 18.015 gram per mole). Here the message is 18.015 grams of water or 18.015 cm³ of water (water density is 1 gram per cm³) contains one Avogadro number of water molecules (Or three times of the Avogadro number of atoms since each water molecule is formed by two H and one O atoms).

The density of any substance on Earth can be quite different. The density of water is 1000 kg/m³, while the density of air is only 1.2 kg/m³. The density of gas hydrogen is even less, 0.082 kg/m³. Nevertheless, one m³ of any substance on Earth will contain a lot of atoms, since the Avogadro's number is huge.

OK, back to the question now. One square inch of the world, it is not sure how deep or how broad in space. To talk about of the total number of atoms, the volume should be correctly specified. As an example, I can give a rough estimate of a square inch of water (one square inch = 6.4516 cm²) with different depths:

1) If it is one meter deep, the volume should be
6.4516 cm² times one meter = 645.16 cm³.

In terms of atom number, it should be
35.81 * 3 * 6.02*10²³ = 6.47*10²⁵

Here we multiply by 3 because each water molecule contains two atoms of hydrogen H and one atom of oxygen O. The number 6.02 times 10²³ is Avogadro's number.

2) If it is only one inch deep, the total number should be less because of the lesser volume. The total number should be scaled by the ratio of one inch to one meter:

6.47*10²⁵ * (one inch/one meter)= 6.47*10²⁵ * 0.0254 = 1.64*10²⁴

3) If the depth is equal to the deepest ocean depth, 10994 meters, then the total number of atoms is about
6.47*10²⁵ * 10994 = 7.11 * 10²⁹

To conclude, the atom number is a huge number in any reasonable amount of substance on Earth. The Avogadro number is the order of atom number that you should refer to. The density can vary from one substance to another, but such difference is small compare to the scale of one Avogadro number.