The footage shows Mr Duterte and his supporters making light of the assault by prison inmates in 1989 when he was town mayor.
He said the female missionary was so beautiful that "the mayor should have been first".
Critics say this shows Mr Duterte is "unfit" for the office of president.
Mr Duterte's supporters however have dismissed the comments as a joke.
The candidate is known for his irreverent speeches and boasts of sexual conquests.
One recent poll had him leading over other contenders ahead of the 9 May election.
The rape comments in a YouTube video are the latest in a string of inflammatory statements - including the proposed mass executions of suspected criminals - made by Mr Duterte.
In the video he refers to the missionary who was murdered while ministering in a prison in the city of Davao in the southern Philippines and goes on to make his remark to laughter at a campaign rally.
President Benigno Aquino's spokesman Herminio Coloma said the comments reflected Mr Duterte's "lack of fitness for the presidency" and his "utter lack of respect for women".
A rival presidential candidate, Vice-President Jejomar Binay, called the comments revolting.
"You are a crazy maniac who doesn't respect women and doesn't deserve to be president," Mr Binay told Mr Duterte in a statement.
Human Rights Watch (HRW) Deputy Asia Director Phil Robertson said the comments "were a disgusting endorsement of sexual violence" for which Mr Duterte should be utterly condemned.
In 2015, HRW described Mr Duterte as the death squad mayor for his tactics in dealing with violent crime in Davao, with more than 1,000 people being killed.

A 26-year-old man sustained facial injuries during the incident on a number 9A bus travelling on Paisley Road, Tradeston, towards Penilee.
The disturbance happened at about 20:00 on 20 December.
Police described the man they are looking for as white, in his 50s, of heavy build. He has blotchy skin and a shaved head.
He was wearing a hooded black body-warmer with a blue zip-top underneath.
Officers have asked anyone who recognises the man or has any further information to contact them.

Coleman senior died in 2014 after his son's first campaign in charge of Wales had ended with the failure to qualify for the World Cup in Brazil.
"I probably helped to put him where he is now," Coleman said before Wales meet Belgium in Lille on Friday evening.
"He loved football and would have loved this. He would have been ecstatic."
Coleman, 46, went on to lead Wales to a first major international tournament for 58 years and even surpassed rivals England in reaching the quarter-final stages of Euro 2016.
"My dad was a huge football fan and unfortunately he witnessed the first campaign," Coleman continued.
After his playing career was cut short by injury, Coleman was appointed Fulham manager in April 2003 and kept the Cottagers in the Premier League in his debut season.
"If you manage your country it is different to managing a club. Your family feel it much more," he added.
"Not getting a result, they feel it. They do. It is a tough one. Equally when it is going great, they are on cloud nine - and at the moment they are loving it."

Temporary closure notices were placed on the shellfish beds after high E.coli bacteria readings last week.
Latest tests have showed satisfactory water quality levels, said the FSA.
Closures remain in place in Devon at Slapton, Taw and Torridge and in Jersey.
An FSA spokesman said: "Recent tests have shown high levels of E.coli contamination in 30 shellfish beds in Cornwall and Devon.
"However, a number of factors have led us to question circumstances which may have led to these unusual results, including the unprecedented nature of the results, the lack of a clear cause and the fact that other testing does not support the view that there is an obvious risk to public health.
"Because of this, the FSA will not follow its normal protocol and in this case will allow the closed shellfish beds to re-open as long as there is one further test that shows the bed has E coli within accepted tolerance levels."
A video of leading Philippines presidential candidate Rodrigo Duterte joking about a murdered Australian rape victim has provoked a storm of protest.
[NEXT_CONCEPT]
Police have released CCTV images of a man they want to speak to following a racist assault on a Glasgow bus.
[NEXT_CONCEPT]
Chris Coleman wishes his father Paddy was alive to see Wales reach the quarter-finals of the European Championships.
[NEXT_CONCEPT]
Shellfish farms on the Helford River and Camel Estuary in Cornwall have re-opened after pollution closure notices were lifted by the Food Standards Agency (FSA).