What is the religious ruling on being in close proximity to women during tawaf in Umrah, and how can one atone for this action?

You erred by committing a sin in the Haram (sacred precinct). Deliberately sinning there is not like sinning elsewhere, as Allah Almighty says: "And whoever intends therein [to deviate] with wrongdoing - We will make him taste of a painful punishment." You must offer sincere repentance and feel deep remorse. If your repentance is truthful, Allah will accept it. Increase your acts of obedience. No specific expiation is required of you, and your 'Umrah is valid as long as the act occurred after it.