Month long Southern Italy + Greece itinerary?

You nailed it: The weather is highly likely to range from hot to miserably hot. I would not want to spend a month-long July vacation in those places. It will be a mostly-outdoor trip, and the heat will be draining. Obviously, a lot of beach time would help.

Sicily for me is a 2- to 3-week destination all on its own. Unless your sole interest is the Greek sites and you plan to see nothing else (maybe 6 or 7 days in a rental car?), I urge you to drop Sicily from the itinerary. Or drop Greece and keep Sicily. But July is a bad time of year to go to Sicily; I wilted badly in June, and I'm a southerner.

What acraven said, plus . . . Not just hot, but HUMID. And of course, crowded.

Even if you're from Houston or Miami, consider whether you'd want to be out and about all day, every day for a month.

I don’t think July is a bad time to go provided you make it mostly a beach vacation with plenty of swimming time.
One month is a decent amount of time, but you are putting a lot of meat on the fire. So you may want to consider dropping one of the 3 localities. If not I suggest at least 10 nights in Sicily, at least 10 in Greece, and at least one week in the Gulf of Naples area.
I’ll give you my proposed itinerary for Sicily. I’m avoiding the interior to avoid extreme heat. The coastal areas have a nice breeze in summer so they are not too bad.
Sicily is big, so you need multiple bases. At that time of the year I would rent a car, because traveling by bus in summer may be miserable.
To visit Palermo (3+ nights) I would base in Mondello, on the beach. Then I would move to Scopello, in between swim time at the Zingaro Reserve you can take short day trips to Erice, Segesta, Selinunte, etc.
Then I would move to near Agrigento and swim on the Scala dei Turchi (Stairs of the Turks). Find time for a day trip to the valley of the temples (it will be hot there). Move then to Siracusa area. Stay nearby at a beach location. The Agriturismo “Masseria sul Mare” near Avola has its own beach. Otherwise find a place at Fontane Bianche, a popular beach resort town. Obviously find time to visit Siracusa and Noto in between beach time. Next move to Giardini Naxos, another beach town. In between swimming time, find the time to visit Taormina, Mt. Etna, Savoca.
Finally move back to Palermo, but on the way stop In Cefalu, another town with a beautiful beach. They have a nice cathedral, try to visit it but not in a bikini.

I've been to all those places in July - multiple times. In fact I've traveled to Europe (mostly southern Europe) every July for most of the last 20 years. I do not think it is too hot to be enjoyable. Some people are heat averse. I happen to be rain averse. I'll take 90 and sunny every day over 75 and rainy. But other people are the opposite. The places you are going are all coastal so there is usually a bit of breeze. Mornings and evenings are delightful. Plan for a 'siesta' mid afternoon back in your air conditioned hotel, or beach time. Or even just a long lunch somewhere cool during the hottest part of the day (about 2-5pm).

Most people (myself included) feel you really need close to 2 weeks to get the most out of Sicily. But if you did that you would not have enough time for all the other places. You could do a week Sicily (would have to pick and choose just part of it), a week Amalfi/Naples, a week mainland Greece, and a week on 1-2 Greek islands. Or you could decide to save Sicily for sometime when you have more time and spread the time out over the other three.

I’ve been in Greece only in June, and although Athens is hot, the islands get a nice breeze and in the evening you even need a sweater. You may not need a sweater in July, but I think he islands will be very enjoyable in July. Obviously in Athens, in the city streets, it will be sweltering hot, especially near the sunny ruins of the Parthenon where there is no shade. Just avoid the hot midday hours for touring the city. Enjoy time in an estiatorio (restaurant) at lunch. Also I would minimize time in Athens to no more than 3 days. I would concentrate on islands and coastal areas also.

For Amalfi coast area, you are on the sea also, so if it’s hot get in the water. Take a boat excursion to Capri too (several depart from Positano for example). There are plenty of beaches on the Amalfi coast, although more than beaches they are cliffs and pebbly coves.

Obviously you can’t go to that area and not visit Pompeii. Walking through those ruins in the middle of the day will be miserable. Go early in the morn, so that you can be done by lunch time when you can head back into the water on the Amalfi coast or enjoy some great fish restaurant.

You could fly into Naples 2 nights. On to Sorrento or the Amalfi coast 5 nights. Back to Naples fly into Catania go straight to Taormina for three nights then two nights Ortegia. From Ortegia go to Ragusa for three nights. Then one night Agrigento and end your Sicily portion with four nights in Palermo. Fly from Palermo to a Greek island of choice for three/4 nights then fly to Athens for your mainland Greece 5 nights.