http://www.naval-history.net/WW2CampaignsRNMed3.htm has the Allied Order of Battle, and it isn't terribly encouraging for the Italians. 4 or 6 Battleships (
Nelson,
Rodney,
Warspite and
Valiant are named, and the map only shows 4 but the OrBat lists "6"), 2 Fleet (
Formidable and
Indomitable) plus one Light and four Escort Carriers, 15 Cruisers, 128 Destroyers and 26 Submarines - not to mention the obvious land-based air from Malta and North Africa.
On the Italian side
Vittoria Veneto was badly damaged on the 5th of June 1943 when she was hit by an American air raid on La Spezia which caused "serious structural damage" forcing her to be shifted to Genoa for repairs.
Roma was hit in the same series of raids but her damage was less severe - however she was under repair from the 1st of July to the 13th of August. Getting her out of dockyard hands after they've already been working on her for a week isn't realistically going to be much quicker than finishing the repair - but by that time the battle for Sicily is pretty much over.
Which means the correlation is
Littorio .vs.
Nelson,
Rodney,
Valiant and
Warspite. And that means you don't have a battle -
Littorio can steam at 30 kts, while the Queen Elizabeths could once have done 24 kts and the NelRods 23 kts. Not being a moron, as soon as the captain of of
Littorio realises that he's pitching his 9 x 15" guns against 16 x 15" and 18 x 16" he'll be running like hell for home and screaming for air cover.