Asked whether he thought homeopathy could cure Ebola, Mr Browning said: "It's not for me to go down that track at all. The World Health Organisation, world health authorities are doing that."
"They will be considering I hope absolutely every possible options to this very concerning disease."
Asked whether that should include homeopathy, he said "Why not?"
"Internationally homeopathy is considered in some places.. I am not an expert but I assume they will look at that as much as a number of other options."
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Homeopathy is a controversial alternative form of medicine which involves dosing patients with extremely diluted solutions of substances linked to the ailment being treated.
In her petition, Ms Sheffield says it has "a proven track record of treating and preventing serious epidemic diseases".
"It's used by governments for dengue fever, leptospirosis, epidemic fever, malaria, and Japanese encephalitis epidemics, and, historically, for other serious contagious diseases.
"The appropriate homeopathic medicine (remedy) is likely to be just as effective against the Ebola virus."
Green Co-leader Russel Norman has distanced himself from the issue, saying he didn't believe homeopathy would be effective against Ebola.
"There are many New Zealanders who use homeopathy but even they would say it's not the right thing to use for Ebola."
"It's not something we support, it's not Green Party policy."
He said he'd spoken with Mr Browning about his signing of the petition.
"He can see that it wasn't such a great idea to have signed that petition and we were certainly very clear with him that we didn't think it was a good idea."
"The Green Party takes an evidence based approach and leaves it up to the WHO about dealing with issues like Ebola."
Dr Norman said he understood Mr Browning's apparent encouragement to others to sign the petition was automatically generated after he signed it, "and he took that down later".
Prime Minister John Key said he thought Mr Browning's actions undermined the global issue of Ebola.
"To sort of argue some homeopathic drug is actually going to solve your issues if you have Ebola is either misguided or taking the mickey."
The petition in support of homeopathic treatment for Ebola:
We call on those within WHO in positions of authority and influence to:
1. Look at homeopathy's record in the treatment and prevention of serious epidemic disease
2. Engage with qualified representatives from the homeopathic profession to formally identify the best-indicated remedies
3. Test those remedies to determine which are best for the treatment and prevention of Ebola
4. Obtain supplies of those remedies from waiting and concerned homeopathic pharmacies
5. Bring the outbreaks under control by distributing these remedies as quickly as possible throughout the affected areas.
Finally, please end the suffering of those in the Ebola crisis by using the tried and proven homeopathic option.