It's important to note before we set off that Gildas' actual history of the fall of Britannia to the Anglo-Saxons, which should mention a historical Arthur if he existed, makes no mention of him but does recount the withdrawal of Roman military and civil support and the slow decline of the province as the Romano-British authorities desperately try to hold things together.
Most of the Arthurian myth as we recognise it today comes from Mallory's Le Morte d'Arthur, written in 1469, but it takes its source material from Geoffrey of Monmouth's Historia Regum Britanniae or History of the Kings of Britain. A plethora of translations and analyses exist; I use Thorpe's 1966 version. To say that Geoffrey of Monmouth is economical with the truth would be like saying that the First Crusade was a light stroll that got a bit carried away, but, hoo boy, does Geoffrey mention Rome.
Following his coronation by 'The Archbishop of the City of the Legions', Arthur sets about driving the Saxons out of England before indulging in a little light genocide, conquering the Irish through starvation and 'wiping out' the Picts so that the Scots begged to surrender and convert. Following this 'restoration of the country to its dignity', Arthur marries Guinevere in York, who is said to come from 'a fine Roman family.'
After conquering Iceland and Norway, Arthur decides to invade 'Gaul', which Geoffrey implies is a semi-autonomous Roman province, ruled by a Tribune, Frollo, on behalf of the Emperor Leo. Arthur supposedly takes 9 years to subdue Gaul, before returning to Britain. The 'Procurator' of Rome, Lucius, then writes a letter to Arthur, condemning his tyranny and for capturing the province of Gaul from the Empire, and demands tribute. Arthur then delivers a speech claiming that, if the Romans can claim tribute because Julius Caesar once invaded Britain, then he too can claim tribute because Belinus once captured Rome and because the Emperor Constantine had been king in Britain. He then invades Rome, defeating a vast Roman army at Saussy, where the Emperor Lucius is killed. Arthur is on the verge of taking Rome itself when news reaches him of his betrayal by Mordred and he returns to Britain.
Rome, therefore, plays an important role in the original sources of the Arthur mythos, although much of his exploits is sanitised by the time of Mallory's writing.
E: As has been pointed out, Mallory does include Arthur's war with Rome.