They do touch, actually.
How do you define “touch”? I’d say “to be close enough to feel the other object pushing back”. That’s what electrons do for atoms – when they get close enough they repel each other.
When you’re holding an object it’s the electrons in your hands pushing against electrons in the object via electromagnetic interactions.
When the atoms in your finger get close to the atoms in an object, the atoms repel each other. This repelling force stops the atoms from getting any closer together; it’s what stops you from being able to put your finger through the object. This force is what you feel when you use your sense of touch. Depending on what the surface of the object is like, the atoms will be arranged in a different way, so the repelling force will repel your finger differently, which you interpret as a different texture.
Basically, when you “touch” something, that really means you are putting your finger atoms close enough that they are affected by the presence of the object atoms, and your sense of touch is how your brain processes how the object atoms affect your finger atoms.
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