Arnold's promise was to not come into the house unless Connie tried to use the phone. So in a way, the thing he does not promise is to absolutely not come in the house. Arnold's convoluted "promise" is really no promise at all; it is simply part of his attempt to persuade Connie to come out of the house and to assure her that he is "the one" for her.
Much of the story is given over to Arnold's rhetoric. His constant reassurances and sweet talk are meant to undermine her confidence. Arnold shows that he knows all about her—that Connie doesn't have any information that might help her resist Arnold. For instance, Arnold knows that her family will not be back; he knows all the details of where they've gone, so Connie's threat that her father might return is hollow.
In fact, while Arnold tries to use his knowledge of Connie and her friends and family as an example of his fascination with her, his real aim is to show his power over her. He mentions the will do "just two things, maybe three" to her, after which he says she will like him the way she likes people she is "close" to. So in another sense, what Arnold does not promise is to not harm her.