He didn't ban the entire Navy, just the Secretary of Navy from being on the Top Policy Group. The bomb project as a whole was kept very compartmentalized, but there were members of the Navy who knew about it (Admiral Leahy for example, and some of the research work was done by the Navy).

As for why he excluded the Secretary of the Navy early on... we don't know. Roosevelt had mysterious ways and liked to keep secrets. Vannevar Bush was present at the discussion about the formation of the Top Policy Group, and he later used to incident in his memoirs to give an illustration of Roosevelt's curious indecipherability. After Roosevelt had named several people to be on the group, Bush asked whether Secretary of the Navy Knox should be included.

He [Roosevelt] looked at me with one of his strange smiles and said, “No, I guess not, not now.”

And that was that. Vannevar Bush, Pieces of the action (New York: Morrow, 1970), 292.

On Roosevelt's predilections towards secrecy in general, see John M. Schuessler, “The deception dividend: FDR’s undeclared war,” International Security 34, no. 4 (2010), 133-165. It is one of the things that marks his Presidency in odd ways, including but not limited to the atomic bomb.