If the cypher being used is secure, encrypting a block of 1s or 0s will generate an apparently random stream of 0s and 1s. There will be no way to determine the private key from the output of encrypting a block of known data with the public key.
Determining the private key given the public key in RSA-2048 requires factorizing large blocks of very large numbers which is not practical given current computing hardware. People are working on "smart" attacks that solve the factorization problem from a theoretical math angle instead of a brute force computational approach and there is some consensus that RSA-2048 may be broken in the next few years. A transition to Elliptic Curve Cryptography (ECC) is underway and RSA has never been approved for Suite-A or Suite-B encryption which might lead one to believe that the NSA has known about the weakness of RSA for a long time.