Mr Morrison on Friday said the government worked with a separate medical contractor - International Health and Medical Services - and appropriate medical support was available.
"We don't go around making a big song and dance about it, we just get on and help people and provide the care that is necessary," he told Sky News on Friday.
Mr Morrison said further medical assistance is organised by the federal government as required.
Aid worker organisation Save The Children joined MSF to express concern for the "devastating physical and mental impacts" on children in detention.
The organisation said it witnessed the deteriorating mental and physical health of detainees firsthand, calling on the prime minister to immediately remove children from the island.
Labor leader Bill Shorten said the government should accept New Zealand's offer to resettle detainees on Nauru.
"Five years has been too long to resettle some of the people we have in our regional processing," Mr Shorten told reporters in Melbourne on Friday.
"We support the American deal. I think the government should accept the offer of New Zealand to resettle some of the people on Nauru."
Mr Shorten fell short of rejecting offshore processing, saying Labor is committed to stopping the boats and preventing deaths at sea.
Save The Children's Mat Tinkler said the prime minister should urgently arrange for children and their families to be moved to Australia or an appropriate third country "until an enduring solution can be found".
The Nauruan government said that MSF staff had shown themselves to be "political activists".
"Although MSF claimed to be a partner to Nauru and the Nauruan people, instead of working with us, they conspired against us," the statement issued on Friday said.
"The facilities, care, welfare and homely environment offered to refugees and asylum seekers on Nauru are comparable or better than what other refugees and asylum seekers across the globe receive."
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