There are following fundamental forces:
Gravitation: long distance force
electromagnetic force: Coulomb law
Weak force: is important in beta decay
Strong force: holds nucleons (proton, etc.) together
Since the electrons are kept near the nucleus because of the attractive Coulomb potential, the electromagnetic force decides the size of an atom.
However, in practice, this isn't just the influence of the electromagnetic force which decides the size of the atom. But it plays the leading role.