Good news for your Dad: he won't have any trouble finding low-carb or carb-less meals in Italy. In the cities, I might suggest looking for Tavola Caldas; these are restaurants that are sort of like cafeterias, where you choose from offerings on display, and pay a cashier. This way, he'll be able to choose foods - like proteins, vegetables and soup - on sight.
A little about them (no this is not my website, promise!):
http://www.italiannotebook.com/local-interest/tavola-calda/
Seafood, meat entrees and veggies will be plentiful in other restaurants as well so no worries!
Rome: you mentioned the Vatican? If he doesn't like art, skip the museums and just go for the basilica. For history, he would, of course, enjoy the Colosseum, Forum and Palatine, and I'll suggest the Portico di Ottavia and Teatro Marcello area: there is signage to tell you what you're looking at:
http://www.turismoroma.it/quartieri/ghetto-circomassimo?lang=en
A walk on part of the Via Appia Antica is an excellent day out, with lots of history and attractions to see; catacombs; the ruins of a circus; crumbling tombs; the excavation of a Roman bath...
http://www.parcoappiaantica.it/en/
The Baths of Caracalla are interesting as well:
http://www.060608.it/en/cultura-e-svago/beni-culturali/beni-archeologici/terme-di-caracalla.html
There are some churches with fascinating historic value as they have Roman ruins in their basements. Take a look at San Nicola in Carcere and San Clemente, for starters:
http://www.060608.it/en/cultura-e-svago/luoghi-di-culto-di-interesse-storico-artistico/chiese-cattoliche/san-nicola-in-carcere.html
http://www.basilicasanclemente.com
You could also take an easy and inexpensive day trip to Ostia Antica - which rivals Pompei:
http://www.ostia-antica.org
There's just a TON of history in Rome so no worries there either! Many of these outdoor spaces - like the baths, Appia Antica, Portico di Ottavia, etc, may be busy but not nearly as overrun as the Vatican or some of the other Top 10 attractions so he'll be less bothered by crowds. The Colosseum is, of course, a big draw but a fair amount of tourists skip the Palatine and even the Forums.