The four properties of a primary key
No, from uniqueness you cannot deduce full functional dependency.

I you have a relation (=table) with a candidate key(~primary key) P and additional attributes A,B,... than all attributes depend from P because P is the candidate key, but all atrributes depend from the combined superkey PA too (this means the key containing both P and A), but this is not a full functional dependency because they depend on a subset of attributes of PA, on the attribute P. PA is unique and not null but PA is no candidate key.