Ellipticity of a given pseudo-differential operator is a metric independent condition. By a pseudodifferential operator I mean an operator $P: C^\infty(M)\to C^\infty(M)$ satisfying H\"{o}rmander's asymptotic conditions described in Definition 2.1 of
L. Hormander: Pseudodifferential operators, Comm. Pure Appl. Math., XVIII(1965), 501-517.
The attribute given signifies that we know $Pu$ for any $u\in C^\infty(M)$.
I think the confusion is about how you define the operator $L_s:=(\Delta_g+1)^s$. More importantly, the confusion can be traced to the fact that on a manifold the identification of a smooth function with a distribution is metric dependent.
More precisely, on a manifold there are two different types of objects: generalized functions (belong to the dual of the space of smooth densities) and distributions or generalized densities (belong to the dual of the space of smooth functions).
If we fix a volume form on the manifold then we can identify a distribution with a generalized function, but this identification depends on the choice of volume form. In the Euclidean case we do not pay much attention to this, because we have a God-given volume form. On manifolds one must be more careful. The kernel of an operator acting on functions is a distribution. (An elegant way out of this is to work with $1/2$-densities.)
Returning to the concrete case at hand, for $s<0 $ the integral kernel of this $(1+\Delta_g)^s$ is the function
$$K_s(x,y)=\sum_k (\lambda_k+1)^s \Psi_k(x)\Psi_k(y),$$
where $(\Psi_k)$ is a unitary basis of $L^2(M,g)$ consisting of eigenfunction,
$$ \Delta_g \Psi_k=\lambda_k\Psi_k.$$
If we want to reconstruct the action of $L_s$ on functions we need to think of $K_s$ as a distribution
$$L_su(x) = \int_M K_s(x,y) u(y) d V_g(y) $$
For $s>0$ you set $L_s=(L_{-s})^{-1}$. You see that the definition of $L_s$ depends on $g$. With this definition you do get a pseudo-differential operator which is elliptic.
