Maybe if Arthur had an ounce of Eames' ease with people (he wouldn't go so far as to call it charm), he'd have followed her, but he just didn't understand what he could possibly say that would extract his foot from his mouth. She obviously didn't want to listen to him and Arthur was thinking that because you backed me into a corner and Eames was being his normal self wouldn't fly by way of a good answer to her question.

So Arthur stood and watched Ariadne walk away before turning around and going back into the bar to drown his anger in whiskey.