Some time ago I worked out how to define Kähler differentials and derive their cotangent and conormal exact sequences in any protomodular category with pullbacks and pushouts. Based on the same idea, I think the following argument works to show that a morphism being unramified is equivalent to the vanihing of its module of relative Kähler differentials.

Given a morphism $X\xrightarrow{f} Y$, the category of **Beck modules equipped with $f$-derivations** has for its objects internal abelian group objects $Y\leftarrow M$ in $\mathcal C/Y$ equipped with an additional section $X\xrightarrow{d} M$ of $X\leftarrow M$ such that $d\circ f=0\circ f$ where $Y\xrightarrow{0}M$ is the zero section of the Beck module. The Beck module $Y\leftarrow\Omega_{Y/X}$ of **Kähler differentials** is the initial object of this category.

For example, a Beck module over a ring $R$ is precisely a square-zero split extension $R\leftarrow R\oplus M$ for $M$ an $R$-module, and a $k$-linear derivation amounts to an $k$-linear map $R\xrightarrow{d}M$ such that $d(ab)=d(a)b+ad(b)$ and $d(c)=0$ for $c$ in the image of $k\to R$.

In the case where $\mathcal C$ is a protomodular category (e.g. the category of rings), recall that the forgetful functor from Beck modules to morphisms equipped with a section has a left adjoint. For the category of rings, a pointed object over a ring $R$ is a split extension $R\leftarrow R\oplus A$ for an $R$-algebra $A$, which left adjoint sends to the Beck module $R\oplus (A/A^2)$ (i.e. the left adjoint sends $R$-algebras $A$ to $R$-modules $A/A^2$).

This allows us to generalize the construction of Kähler differentials to arbitrary protomodular categories as follows.

Define a weaker notion of a **pointed object equipped with an $f$-prederivation**, i.e. a morphism $Y\leftarrow P$ equipped with a "point" section $Y\xrightarrow{0}P$ and an additional section $X\xrightarrow{d} P$ of $X\leftarrow P$ such that $d\circ f=0\circ f$.

For example, a pointed object in the category of rings is a split extension $R\leftarrow R\oplus A$ for $A$ an $R$-algebra, and a $k$-linear pre-derivation amounts to an additive map $R\xrightarrow{d} A$ such that $d(ab)=d(a)b+ad(b)+d(a)d(B)$ and $d(c)=0$ for $c$ in the image of $k\to R$.

The pointed object $\Gamma_{Y/X}$ of **Kähler pre-differentials** is then an initial object in the category of morphisms $Y\leftarrow P$ equipped with a pair of sections that make a fork $X\xrightarrow{f}Y\overset d{\underset 0\rightrightarrows} P$.

It turns out that $\Gamma_{Y/X}$ is then given by the cokernel pair of $X\xrightarrow{f}Y$, and $\Omega_{Y/X}$ by its reflection in Beck modules. Hence in the category of rings, $\Gamma_{R/k}$ is $R\leftarrow R\otimes_kR$ and $\Omega_{R/k}$ is $R\leftarrow R\oplus(J/J^2)$ where $J=\ker(R\leftarrow R\otimes_k R)$.

Recall that in a protomodular category, being an abelian group object is a property of pointed objects, rather than a structure. Accordingly, we define a morphism $A'\xrightarrow{f} A$ in a protomodular category $\mathcal C$ to be a **square-zero morphism** if its kernel pair $K[f]\rightrightarrows A'$ equipped with the diagonal morphism $A'\xrightarrow{\Delta}K[f]$ is a Beck module over $A'$.

For example, $Y\leftarrow\Omega_{Y/X}$ is a square-zero morphism because its kernel pair is the pullback of a Beck module so still a Beck module.

Thus, we can define a morphism $X\to Y$ to be **unramified**, respectively **etale**, respectively **smooth**, if any square
$\require{amsCD}
\begin{CD}
X @>>> A'\\
@VVV @VVV\\
Y@>>> A
\end{CD}$
can be filled so that the resulting diagram commutes with at most one, respectively at least one, respectively exactly one morphism $Y\to A$.

We can now easily show that unramified is equivalent to $\Omega_{Y/X}=0$, i.e. to the fact that the zero section $Y\xrightarrow{0}\Omega_{Y/X}$ and the canonical $f$-derivation $Y\xrightarrow{d}\Omega_{Y/X}$ are equal.

First, unramified implies $\Omega_{Y/X}=0$ since there must be at most one fill in the square
$\require{amsCD}
\begin{CD}
X @>>> \Omega_{Y/X}\\
@VVV @VVV\\
Y@= Y
\end{CD}$
and these fills are precisely the $f$-derivations $Y\to\Omega_{Y/X}$.

Second, here's a sketch of a proof that $\Omega_{Y/X}=0$ implies $X\to Y$ is unramified. The pullback of the square-zero $A\leftarrow A'$ along $Y\to A$ is still square-zero, hence it suffices to show that there is at most one fill in any square
$\require{amsCD}
\begin{CD}
X @>>> A\\
@VVV @VfVV\\
Y@= Y
\end{CD}$. But a pair of fills $Y\overset a{\underset b\rightrightarrows}A$ is the data of a morphism $\Gamma_{Y/X}\to K[f]$ of pointed objects over $Y$. Since $A\xrightarrow{f}Y$ is square-zero, $K[f]$ is by assumption a Beck module, hence $\Gamma_{Y/X}\to K[f]$ factors as $\Gamma_{Y/X}\to\Omega_{Y/X}\to K[f]$. Finally, triviality of $\Omega_{Y/X}$ forces uniqueness of the morphism $\Omega_{Y/X}\to K[f]$.