Let me illustrate my question by starting with the simplest possible example: Let us consider $P := - \mathrm{d}^2/\mathrm{d}x^2$, an elliptic partial differential operator on $\mathbb{R}$; let us also consider the following boundary-value problem on the interval $\overline{\Omega} = [0,1]$: \begin{equation} P u = f, \qquad u(0)=u(1)=0. \end{equation} As is (I think) well-known, when seen as an operator $L^2(\Omega) \to L^2(\Omega)$, $P$ is unbounded. However, it is closed on the dense domain $D(P) := H^2(\Omega) \cap H_0^1(\Omega)$ where $H_0^1(\Omega)$ is the closure of $C_{\mathrm{c}}^\infty(\Omega)$ in the $H^1$ norm (so that any element of this space has vanishing trace on $\partial \Omega = \{0,1\}$, i.e. it satisfies the Dirichlet boundary condition above in a weak sense). Furthermore, $0$ is in the resolvent of $(P,D(P))$, i.e. there exists a bounded inverse $P^{-1} : L^2(\Omega) \to L^2(\Omega)$. In fact, in this example the inverse is easily computed: it is the integral operator defined by the (continuous, as it happens) kernel \begin{equation} G(x,y) = \begin{cases} x(1-y) & x \leq y \\ y(1-x) &x > y \end{cases}, \quad (x,y) \in \Omega \times \Omega. \end{equation} Of course, when viewed as a distribution in $\mathscr{D}'(\Omega \times \Omega)$, $G$ is the Schwartz kernel of $P^{-1}$ which we know on abstract grounds must exist since $P^{-1} : C_{\mathrm{c}}^{\infty}(\Omega) \to \mathscr{D}'(\Omega)$ is continuous.
My question is the following: in this example and in more general examples where $P$ is a second-order elliptic differential operator on, say, an open (and not necessarily compact) region $\Omega$ with smooth boundary in $\mathbb{R}^n$, and assuming that we can find a suitable dense domain $D(P)$ for $P$ as above so that $(P,D(P))$ has a bounded inverse $P^{-1} : L^2(\Omega) \to L^2(\Omega)$, does the Schwartz kernel $G$ of $P^{-1}$ define a pseudodifferential operator on $\Omega$?
I'm by no means an expert on the topic, but it seems to me that you are looking for Boutet de Monvel's calculus. In this calculus you can construct parametrices of pseudodifferential operators with boundary conditions. So, in short the answer is yes and the remnant you get is a so-called singular Greens operator which is smoothing in the interior, but acts as a psdo on the boundary. A good reference is the book by Grubb (there is also an article by Schrohe (short introduction to ..) which gives an overview of the calculus, but no applications).