**No**, for an elliptic curve group as used in cryptography, it is not possible to determine from the coordinates $(Q_x,Q_y)$ of public key $Q$ if in it's evaluation as $dG$, the last operation was a point doubling or a point addition.

One fundamental issue is that what's asked is unspecified, because the point multiplication algorithm used to compute $dG$ from $d$ is unspecified. Several algorithms for that yielding the same $(Q_x,Q_y)$ exist, and they differ in regard to if the final operation is a point doubling or a point addition.

In particular, one of the simplest algorithm to compute $dG$ from $d$ is:

$R:=\mathcal O$

$S:=G$

For each bit $d_i$ of the binary expression of $d$, starting from low-order bit

- if $d_i=1$ then $R:=R+S$
- $S:=2S$

Output $R$.

For every valid $d$ (that is $0<d<n$ where $n$ is the order of $G$), the operation $R+S$ that is conditionally performed is a true point addition (that is $R\ne S$). It follows that the final output always is obtained by point addition.

Even if we specify an algorithm such that the nature of the last operation is determined by $d\bmod2=d_0$, we believe that it's not computationally possible to guess that bit $d_0$. Argument: if that was possible, another algorithm (detailed below) can be constructed that would allow to find the private key $d$. But elliptic curves groups used in cryptography are chosen so that it's believed practically impossible to find $d$ from $Q$ by any existing mean.

Assume there's an algorithm that, from the coordinates $(Q_x,Q_y)$ of public key $Q$ such that $Q=dG$ with $0<d<n$ (where $n$ is the prime order of $G$), outputs bit $d_0$ of $d$.

Given $Q$ as $(Q_x,Q_y)$, test if $Q=G$, in which case $d=1$. Otherwise, compute $d_0$ with the algorithm, then $(Q'_x,Q'_y)$ for point $Q':=((n+1)/2)(Q-d_0G)$. It holds $Q'=d'G$ with $0<d'=\lfloor d/2\rfloor<n/2<n$, therefore the algorithm can be used to find the low-order bit $d'_0$ of $d'$, and that is the second-low-order bit $d_1$ of $d$.

That can be extended to find all bits of $d$, and to a probabilistic algorithm that works even if the hypothesized algorithm finds $d_0$ only for a (distinguishable, non-vanishing) subset of the $Q=dG$, and is not entirely reliable.