Not only you can in theory, but also in practice I suspect both Gavin and Jeff would actually welcome multiple independent implementations of the protocol, since that's a good way for any potential vulnerabilities to be found and fixed.
In other words, go forward!
Yes, diversity is good. If you do decide to go for it, do lots of testing on either the test network or with a testnet-in-a-box setup before even THINKING about handling real bitcoins. If you screw up and lose other people's money it will take a long time to earn back their trust.In other words, go forward!
Oh, and speaking of trust... unless you have a recognized Brand Name, I don't think you'll have any success with a closed-source client.