Is ECB mode safe to use with RSA encryption?
It is highly misleading to call how RSA Encryption is used as 'ECB mode'.

With ECB mode, we break the plaintext into N bit segments, and send each one through the block cipher separately.  The block cipher is deterministic, and so if two plaintext blocks happen to be the same, so will the corresponding ciphertext blocks.

Now, with RSA encryption, we take the short message, and some randomness, stir them together (using a padding method), and send that through the RSA primitive, resulting in the ciphertext.  If we encrypt the same message again, well, we'll stir in different randomness, resulting in a different looking ciphertext.  Hence, we avoid the problems inherent with ECB mode (and thus it is rather inappropriate to call what we do with RSA 'ECB mode').

Now, some closing comments:


The above assumes that you are using a known-good RSA padding method, such as OAEP.  If you aren't, and are instead using 'textbook RSA' (which is just to take the message, interpret it as a large integer, and send that to the RSA primitive), then there are a number of security problems you can run into, in addition to determinism.  If that is the case, you definitely need to fix it.
Someone could devise a CBC-type mode for doing RSA encryption (by breaking up the message into separate blocks, and doing an "add mod N" of the ciphertext of one block to the plaintext of the next).  We never do so, because of efficiency - if we ever need to RSA encrypt a long message, a far more efficient approach would be to pick a random (say) AES key, RSA encrypt the AES key, and then AES encrypt the actual message with the AES key.  Because AES is far more efficient than RSA, this is faster (and we retain the only-someone-with-the-private-key-can-decrypt property of pure RSA).