The page was created last year and had sparked objections from members of the armed forces and the public.
The site initially left it in place, saying it did not breach its rules.
However, Facebook removed the page shortly after being contacted by the BBC, saying a check had revealed the account holder's details to be fake.
Critics of the company have said the case highlights concerns about its review policies.Community standards
The page was created last July and had called on visitors to "support the cause in weeding out and eliminating this worthless breed of cowardice".
Facebook's Community Standards state that it will remove content where it perceives there to be a "genuine risk of physical harm" and that members may not "credibly threaten others, or organise acts of real-world violence".Some Facebook users warn that hate pages can cause tension offline
However, a spokesman for the social network indicated that the threat had not been specific enough for its complaints team to act on.
"Sometimes there is content on Facebook that expresses angry and unpleasant ideas but doesn't directly target anyone," he said.
"In such cases the page may be left up. However, we can compel people who post things like this to make their real names visible so they are publicly accountable for their views.
"On investigating this particular page administrator, we found they were using a fake account and we removed it."Continue reading the main story “Start Quote
Clearly this page was taken down because it was offensive”End Quote Dr Joss Wright Oxford Internet Institute
He confirmed that the fact the creator's details were false was only flagged after the site was contacted by the BBC's technology correspondent Rory Cellan-Jones, who had asked about the matter after it had been raised by a member of the audience.'Show respect'
Among those to have previously complained about the posting were visitors to another page on Facebook set up to support members of the Royal Marine commandos.
Its administrator - who had previously served in the British Army - said that this was just one of several pages concerning the armed forces that they had complained about, and added that he believed Facebook needed to be more proactive about deleting such material despite concerns this might raise about free speech.
"You've got to show respect for the men and women who have fought and died for others to hold their own opinions," Stef Proietti told the BBC.
"It creates hatred and causes anger.
"I appreciate that a lot of these types of pages are set up by what you would call internet trolls looking to create a reaction.
"But, as we've seen in the past... it creates more tension, not just among the social network's community but it also spills out into real life."Mark Zuckerberg's site says it has more than one billion monthly active users
By contrast, La Quadrature du Net (Squaring the Net) - a Paris-based group that campaigns for internet users' rights - said it was concerned that a company with as much influence as Facebook should be left to make such decisions.
"A judge may or may not have considered that this was a direct call to violence, and on that ground may or may not have asked Facebook to remove it - and this is how it should be," said the group's co-founder Jeremie Zimmermann.
"[Instead] Facebook has become a sort of parallel justice with its own rules that we cannot fully understand.
"This is a major problem for whoever believes their speech is protected on Facebook."First Amendment rights
But one academic said it was no surprise that the company had acted the way it had.
"The principle of First Amendment freedom-of-speech rights is something that nobody wants to be seen to be violating in the US, particularly Silicon Valley-type companies for which it's buried very deeply into their ideology," said Dr Joss Wright, of the Oxford Internet Institute.
"Clearly this page was taken down because it was offensive, but it's very convenient for the firm to have an alternative justification - the use of fake credentials or, as we've seen in other examples, a violation of copyright.
"I think Facebook will stick to this kind of approach as long as it can. It doesn't want to be put in a position where it's expected it will police its content because that could then turn into a requirement that is forced upon it."