It is aimed at curbing its capacity to develop a nuclear weapon in return for the lifting of sanctions.
Mr McCully chaired the Security Council session when the deal was passed unanimously last week, and he also chaired a regularly scheduled Security Council debate on the numerous conflicts in the Middle East.
He told the Herald the co-operation on Iran was a sign the Security Council "could do a hell of a lot better" on other challenges.
"There is a bit of tendency in this place to focus on looking for humanitarian support mechanisms, which, of course, is laudable, but without putting enough effort into stopping the fighting in the first place.
"We've actually seen the bar set far too low around this place on preventing conflict."
He said a lot of "air time" was given to the fundraising of humanitarian organisations "but actually we need to put more effort into stopping the fighting, making sure that we don't have the huge humanitarian costs we have seen accumulate in Syria, Yemen, Libya and other places".
Mr McCully said there would be a possibly very narrow window of opportunity between the dust settling on the Iran deal and the US presidential contest to get progress on the Middle East peace process.
"It would be easy for that window to be missed - there would be plenty who would be happy for it to be missed, I guess - and our job is to make sure it is not."