Of course, it depends on how close they come to each other, but "no" is the answer.

That means that they do not form chemical bonds (with a few exceptions). It does not mean that the atoms do not interact with each other.

An atom is the smallest particle of an element that retains the chemical properties of that element.Atoms are electrically neutral, with a positively charged nucleus that binds one or more electrons in motion around it

You're absolutely right that there is no meaningful way to classify an individual atom as a solid, liquid, or gas, as these terms are based on interactions between atoms or molecules.