No, this is not true. A counter-example is Ukraine.
My interpretation of the claim is that:
America is a democracy where it is possible to self-nominate for Presidency, without having to get permission or pre-selection from existing parties, cartels or government bodies as part of the normal administrative process for candidacy, AND
There is no other country with a democracy that has that property, for Head of State or Head of Government.
From discussion on the question and other answers, I see that some people challenge the former of these propositions. I am ignoring that and targeting the latter.
The President of Ukraine is:
elected by the citizens of Ukraine on the basis of universal, equal and direct suffrage by means of secret ballot for five years.
In 2014, Intefax reported:
a total of 23 candidates will take part in the presidential race, seven of them are nominated by political parties, 16 are self-nominees.
So, the electoral system support self-nomination, and everyone votes for the candidates directly.
Note that nomination requires the collection of massive numbers of signatures. I consider this to be merely administrative evidence that the potential candidate is a serious contender, as opposed to getting permission from a party, so I argue this still fits into the definition.
I understand Russia to have a similar signature-gathering hurdle. I suspect there are many more examples.
So the USA is not the only democracy in the world where anyone can declare himself or herself a candidate for the presidency.