Diffraction through an aperture gives a half-angle divergence of $1.22\lambda/a$ where a is the aperture diameter. The half-angle divergence of a Gaussian beam is $\lambda/\pi w_0$ where $w_0$ is the $1/e^2$ beam waist.
Are the output aperture of a laser and the beam waist the same things in this instance, ie $a=w_0$? Then why don't the two expressions for divergence equate, ie $1.22 \ne 1/\pi$
Can you equate the two half-angle equations to get the $1/e^2$ beam waist: $w_0=a/1.22\pi$