Does Proj induce some equivalence of categories involving graded rings?

This question is over a year old, but I have a different perspective which might interest you.

As others have pointed out, $\text{Proj}$ is not even a functor, let alone fully faithful, but it does factor through an equivalence in the following sense. A $\mathbb{Z}$-grading on a ring $A$ is the same as an action of the multiplicative group $\mathbb{G}_m$ on $\text{Spec } A$. Homomorphisms of graded rings correspond to $\mathbb{G}_m$-equivariant morphisms of schemes. So $\text{Spec}$ induces an equivalence from graded rings to affine schemes with $\mathbb{G}_m$-action.

But $\text{Proj}$ is only defined on nonnegatively graded rings. Call a $\mathbb{G}_m$-action on scheme $X$ contracting if it extends to an action of the multiplicative monoid $\mathbb{A}^1$. Intuitively, this means that the $\mathbb{G}_m$-action contracts $X$ onto its fixed point locus $X^{\mathbb{G}_m}$. Then a graded ring $A$ is nonnegatively graded if and only if the corresponding $\mathbb{G}_m$-action on $\text{Spec } A$ is contracting.

So we can interpret $\text{Proj}$ as a construction which builds a scheme from an affine scheme with contracting $\mathbb{G}_m$-action. Namely, it sends $X$ to the GIT quotient $(X \setminus X^{\mathbb{G}_m})//\mathbb{G}_m$.

The failure of functoriality is due to the fact that a morphism $X \to Y$ can send points in $X \setminus X^{\mathbb{G}_m}$ to points in $Y^{\mathbb{G}_m}$. There is also some loss of information because any nonzero power of the $\mathbb{G}_m$-action produces the same GIT quotient: this is the Veronese stuff.

Unfortunately, there is not. As Mariano points out, non-isomorphic rings might happen to have isomorphic Proj. However, the biggest problem is pointed out by Alex: Proj is not even a functor, unless you add some restrictions. The problem is already illustrated at the level of vector spaces. If $V \to W$ is a map of vector spaces, you do not get a map $P(V) \to P(W)$ from it, unless it is an injection. Indeed, where would you send lines in the kernel?

I'm not sure about proj but if you think about not just rings but modules over those rings then you get some results. First let us look at affine schemes. We have the usual $$\operatorname{Hom}(A,B) = \operatorname{Hom}(\operatorname{Spec} B,\operatorname{Spec}A),$$

i.e. there is an equivalence of categories between affine schemes and commutative unital rings. Now you can ask the following. If I study modules over a ring, is there a "module-theoretic" category to which this is equivalent to? Yes! Your answer is given by the category of quasi-coherent sheaves on a space $X$ denoted $\operatorname{QCoh}(X)$. Now if $X = \operatorname{Spec} R$ is affine, there is an equivalence of categories

$$\textbf{$R$-Mod} \leftrightarrow \textbf{QCoh}(X)$$ where the equivalence is given by $M \mapsto \widetilde{M}$ and for the other way we send a quasi-coherent sheaf to its global sections. One can also prove the analogous (though arrow preserving!) result $$\operatorname{Hom}_R(M,N) = \operatorname{Hom}_{\mathcal{O}_X}(\widetilde{M},\widetilde{N})$$

where the R.H.S. is global Hom and not sheaf hom.

Now onto your question about Proj. I don't know of a result involving purely proj but if you look at "modules over rings" we have:

Exercise 2.5.9 Hartshorne: Let $S$ be a graded ring, generated in degree $1$ as an $S_0$-algebra. Let $M$ be a graded $S$ module and $X = \operatorname{Proj} S$. There is an equivalence of cateogires between the category of f.g. graded $S$-modules (modulo the equivalence that $M\sim M'$ if they agree in sufficiently high degree) and coherent sheaves on $X$.