(This is a repeat of an above comment.) The category of affine schemes is not Mal'cev. This can be disproven by producing an reflexive, non-symmetric relation on an affine scheme $X$ whose graph is a closed subscheme of $X\times X$.

Take $X=\mathbb{A}^1=Spec(\mathbb{C}[x])$. The relation $(x,x)$ and $(x,0)$ (as $x$ runs over all points) is reflexive and asymmetric. If we identify $X\times X$ with
$$Spec(\mathbb{C}[x]\otimes \mathbb{C}[y])=Spec(\mathbb{C}[x,y]),$$
then the graph of the quotient is the union of the lines $x=y$ and $y=0$. This is a closed subscheme. In terms of rings, this looks like the quotient $\mathbb{C}[x,y]\rightarrow \mathbb{C}[x,y]/(xy-y^2)$.

There's also a simpler and less-satisfying example. Since the category of affine schemes contains the category of finite sets (with $[n]\rightarrow Spec(\mathbb{C}^n)$, one can choose any reflexive, non-symmetric relation on a finite set (since they are all algebraic).