I believe that Hashirama's natural occuring Wood Release is from the fact that he has a life-force and vitality that rivals that of the Uzumaki Clan. Which is why there is also a Mud Release, Hashirama's Life-Force allowed him to combine his two affinities of Earth and Water into Wood Release. Whereas others who also have an affinity to both Earth and Water, or even those who are in the Senju Clan do not obtain Wood Release. My theory is that Naruto could have gained it naturally should a condition be met.
Recalling the Kekkei Genkai Lore, Nature Transformation type Kekkei Genkai is formed when a shinobi has an affinity for more than one Nature Element. Example being Mei Terumi where she has an affinity for three elements, as mentioned in the anime. Fire, Water and Earth. Allowing her to have two Kekkei Genkai: Boil Release and Lava Release.
Back on topic. Naruto is part of the Uzumaki Clan, which means in theory he has potentially higher amounts of vitality and life-force than Hashirama. Combine that with the fact that he is also a reincarnation of Asura could give him this Kekkei Genkai as well. The Condition that I mentioned earlier that if met could give him Wood Release, is that if he has an affinity for both Earth and Water as well.