Show that any continuous $f:[0,1] \rightarrow [0,1]$ has a fixed point $\zeta$

Yes, your proof is pretty much correct. I might change it slightly so that it's presented better.

Let $f:[0,1] \rightarrow [0,1]$ be a continuous function. Show that there is a $\zeta \in [0,1]$ with $f(\zeta)=\zeta$ (that is, show that there exists some fixed point $\zeta$).

Consider the function $g:[0,1] \rightarrow [-1,1]$ defined by $g(x):= f(x)-x$. Note that since $f$ is continuous and polynomials are continuous, $g$ is also continuous.

Now observe that since $f(0)\in [0,1] \implies f(0) \ge 0$, we have: $$ g(0) = f(0)-0= f(0) \geq 0 $$ Likewise, since $f(1) \in [0,1] \implies f(1) \leq 1$, we have: $$ g(1)=f(1)-1 \leq 1 - 1 = 0 $$ Thus, since $g(1) \leq 0 \leq g(0)$, we know by IVT that $\exists \zeta \in [0,1]$ such that $g(\zeta)=0 \iff f(\zeta) = \zeta $. Hence, $\zeta$ is a fixed point of $f$, as desired. $\Box$

The basic argument is a correct one, but there is at least one incorrect statment, which is "$g(x)=f(x)-x\iff f(x)=x$" which isn't what you meant. You were trying to define $g$ and then separately say $g(x)=0\iff f(x)=x$.