I'm leery of using monthly averages for much of anything weather-related, and anecdotal reports from others who spent a few days in your target area, perhaps in early rather than late June, are of limited usefulness. Although early and late July may have similar weather patterns, that won't necessarily hold true for June; after all, early June is spring and late June is summer. For a more reliable perspective, I suggest going to wunderground.com and checking the actual day-by-day weather stats for this year, last year, and as far back as you have time to go. If you must use monthly averages, be aware that late June may be roughly at the mid-point between "June" and "July" stats.
I'm familiar with summer in Tucson and currently live in Washington DC. One difference I've observed is that in Arizona, you feel a lot cooler if you move into the shade--I guess because the humidity is low. Here in DC the heat follows you into the shade. And of course 90 F in the sun feels hotter here than the same temperature in Arizona in the sun, again because of the humidity.
I think you would survive Italy in late summer because of where you currently live; I managed last year from late May to late July (including Sicily and Puglia), but the heat was draining. I don't remember having a "comfortable" day after the end of May except for those occasions when I was high up in the Dolomites. One thing to keep in mind is that unlike in the US, you will not find every little Italian café, shop and museum air-conditioned. Museums showing paintings are cooled; other museums, not so much. Small eating establishments and stores may well just have a fan or two, the exception being high-end shops. Not being able to escape the heat anywhere except possibly your hotel room (assuming it is conveniently located and air-conditioned--don't assume!) can make things difficult.
In sum: There is a reason so many Italian establishments close for an extended period during summer afternoons.