What's the most natural way to establish the asymptotics of $\Delta$ on a compact Riemannian manifold $M$ of dimension $N$? The asymptotics should be $$ \#\{v < A^2\} = \mathrm{const}\ast\mathrm{vol}(M)\ast A^n + o(\mathrm{something}) $$ (Perhaps one could consider first the case of a Kähler manifold? The Laplacian is particularly simple there.)

The most natural way is to study the shorttime asymptotics of the heat or wave kernel on M. For example, you can use the heat kernel $p_t(x,y) = \sum_i e^{\lambda_i t} f_i(x) \overline{f_i(y)}$ where $f_i$ are the eigenfunctions with eigenvalues $\lambda_i$. This is a fundamental solution to the heat equation. When $t$ is small then you can construct a good approximation to $p_t$ near any particular $x$ by hand, using Fourier analysis in local coordinates. The end result is that that $p_t(x,x) \approx C t^{n/2}$. Now integrate this estimate $dx$, noting that $\int_M p_t(x,x)dx$ basically counts eigenvalues with $\lambda_i \leq 1/t$. 


Wener Müler  Weyl laws... http://www.math.unibonn.de/people/mueller/papers/weyllaw.pdf has a pretty good introduction. Edit: I see now that only the second section addresses you specific question and not the intro:( 

