Deterministically generate a RSA public/private key pair from a passphrase (with sufficient entropy)?
Yes, it is possible to deterministically generate public/private RSA key pairs from passphrases. For even passable security, the passphrase must be processed by a key-stretching function, such as Scrypt (or the better known but less recommendable PBKDF2), and salt (at least, user id) must enter the key-stretching function; the output can then be used as the seed material for the RSA key generation. This works for any public-key cryptosystem.

Security is equivalent to what you have in e.g. PGP/GPG when assuming an adversary has access to the private key file protected by the (unknown, key-stretched) passphrase. However, there are drawbacks:


one is giving away the considerable degree of security obtained, in normal practice, by not making the  private key file public; that first line of defense is lost, only the (stretched) passphrase remains;
it is impossible to change the passphrase while keeping the same key and the advantage that only the passphrase is needed;
the key pair generation method should remain static; any change will end up with a different key pair; that precludes periodically increasing the passphrase-stretching parameters to take into account hardware progress (while keeping the same key and the advantage that only the passphrase is needed);
as a consequence, there's a very difficult compromise between security for many years with the same public key, and acceptable usage speed now;
as a minor aside, due to variability of RSA key generation time (at least by standard techniques), some passphrases/keys will require more time than others when deciphering or signing.




No, it is not currently possible that a public key of a traditional public-key cryptosystem (not based on communication with some server) can be (re-)generated from something that a typical human is willing to memorize (perhaps, 80 bit worth of entropy, about 24 digits, or 3 phone numbers), much less from a passphrase a typical human can choose and no other information.

In RSA with $n$-bit modulus, we can reduce the public key to about $n/2$ bits with a good argument that it does not reduce security, or to $n/3$ bits with no argument that it reduces security (see this question), but apparently not much further. Other public-key cryptosystem (like ECDSA) have a more compact public key for equivalent security, but it is still several times above that 80-bit limits, thus impractical to memorize, and even painful to key-in from paper. I have strong doubts that we can ever have a reasonably secure public-key cryptosystem with a public key less than 160 bits.