Fallon, the first senior cabinet minister to visit the islands in 14 years, said that the islands have moved on since the war in 1982, but that Corbyn hadn't.
The defence secretary makes the trip after former Argentine President Cristina Kirchner, who repeatedly claimed that the islands should be returned to Argentina with increasing rhetoric, left office, pathing the way for an easing of relations.
Read more: Jeremy Corbyn wants to hand back Falklands
"The biggest threat at the moment isn't Argentina, it is Jeremy Corbyn and Labour party who want to override the wishes of the islanders," Fallon said.
He added that Corbyn had made it clear the islanders "shouldn't have a veto over their future" and as such posed an "immediate threat".
Three years ago, the people of the Falklands voted overwhelmingly in favour of remaining a UK overseas territory.
However, in January outgoing Argentine ambassador to the UK Alicia Castro claimed that Corbyn wanted to hand back some control of the Falklands to Argentina.
Argentina has maintained Britain has occupied the islands illegally since 1833.
A senior member of the Falklands' government has however hit back at Fallon's claims. "Corbyn has his views. We've been advising him to listen to Falkland islanders so he properly understands modern developments to our way of life. But I don't think he's a threat," Michael Summers told the BBC.
Fallon also confirmed the Ministry of Defence would spend £180m on improving the islands' defences over the next decade.