Is there a way to determine the private key (or the phi value) without $n$ factorization if one knows the ciphertext and the public key and that each letter of the (English) alphabet has been encrypted individually?

**No**, for proper choice of public and private key.

Argument: if one knows $n$, $e$, and a matching private exponent $d$, then one can factor $n$ with little effort. Same if one knows $n$ and $\varphi(n)$. See this for how. Hence, if what's asked was possible in general, it would not be much harder to factor $n$, which what's asked excludes.

Alternate argument: for like half a century, researchers have tried and failed to find how to factor the public modulus $n$ (or find a working $d$) by putting to good use plaintext/ciphertext pairs for textbook RSA; that's even if they are allowed to iteratively choose the plaintext or the ciphertext.

As an aside, in many definitions of RSA, there are several equivalent private keys, and no way to tell which is **the** private key.

On the other hand, if $n$ is small enough that it can be factored, or if there is a small $d$ that works, then a working $d$ can be found (computed by one of the normal methods used in RSA, or by enumeration of small $d$ and checking them against the available plaintext/ciphertext pairs). A conclusion of that is: discussing security of RSA with poor choice of key (including, $n$ or $d$ less than some hundred decimal digits) is pointless.