Bruce Wayne isn't even the only reasonably wealthy, tall, white, buff male in Gotham or adjacent areas. It does not seem obvious to me that Bruce Wayne is Batman, but it does seem solvable given time. Many people here are saying it wouldn't be solved because "my first thought would be..". But your first thought is wrong, so when that doesn't pan out, then what? They'll try something else. It's possible Bruce wouldn't be outed but given time there are clues leading to him, and once you have the inkling that Bruce is Batman, it is fairly obvious from there (obvious meaning it will definitely be figured out conclusively).
By the way, this is a bit off topic, but anyone else notice that the way "Robin" determined Bruce's identity in TDKR was very similar to how Cassandra Cain did? I didn't notice that until yesterday when I was rereading Batgirl.