"Isis (IS) are Sunni, and the fact is that in recent days Isis have not been killing the Christians and the Yazidis and the Shia, they've been killing their own Sunni; the Sunni that do not support them," said Canon White to Premier's News Hour.
On Sunday government forces abandoned their positions, reportedly making a chaotic retreat after days of intense fighting, leaving many of its population to flee for their lives.
Around 500 people on both sides have died during intense fighting in the last two days, including policemen who had run out of ammunition, and civilians caught in the crossfire.
Some 8,000 people have been displaced over the same period, according to the International Organisation for Migration.
An online video showed Humvees, trucks and other equipment speeding out of Ramadi, with soldiers gripping onto their sides.
"Ramadi has fallen," said Muhannad Haimour, a spokesman for the governor of Anbar province. "The city was completely taken. ... The military is fleeing."
Ramadi is the capital of Iraq's largest province, Anbar, and is just 70 miles west of Baghdad.
A statement thought to be from Islamic State (IS) said its fighters had "purged the entire city." It said IS had taken the 8th Brigade army base, along with tanks and missile launchers left behind by troops.
A "very well-placed source" in the Anbar governor's office told the BBC Ramadi was now under the full control of Islamic State, and all government troops had withdrawn.
Prime Minister Haider al-Abadi had ordered troops to stand their ground, saying he was deploying Shia militia to the city.
But an army officer told the BBC that most troops had retreated to a military base in the city of Khalidiya, east of Ramadi.
Anbar province covers a vast stretch of the country west from the capital Baghdad to the Syrian border, and contains key roads that link Iraq to both Syria and Jordan.
IS reportedly controls more than half of Anbar's territory.
Vicar of Baghdad, Canon Andrew White speaking to Premier's Des Busteed:
