Prove that the unit circle cannot be a retract of $\mathbb{R}^2$-Munkres sec 35 exercise 4

The result that no such retract exists is equivalent to Brouwer's fixed point theorem (BFPT) for $n=2$. So, if you want to prove it without any algebraic topology, you have to look at proofs of BFPT which do not use algebraic topology. These do exist. There is one using Sperner's lemma on colouring triangle vertices (see Proofs from the Book) and a really nice one using the fact that the game of Hex can never be a draw (see article by David Gale). There are calculus ones, but I'm not familiar with them.

A retract of a contractible space is contractible.

But $S^1$ is not contractible.

If you don't know about the fundamental group, or Brouwer's fixed point theorem, I still think there's an equivalent formulation in terms of extensions of maps from $S^1$ to the disc. It's equivalent to Brouwer, but says there's no extension of the identity function of $S^1$.