What is the shape of an antennas output?
You can express the solution to the electromagnetic wave equation using various different basis functions. Usually we choose a basis set that is convenient for matching up some boundary conditions that apply to our problem. The gaussian beams are a convenient basis set when working with a beam emitted from an aperture. But you might prefer the spherical harmonics if your antenna radiates more isotropically, for example. 

TLDR: Gaussian beams are just one choice of how to express the emission from an antenna.
The gaussian beams do diverge with an angle, which for the 0th order beam is given by 
$$\Theta = \frac{\lambda}{\pi w_0}$$
where \$\Theta\$ is the full angle of divergence (not the half-angle), and \$w_0\$ is the beam's waist diameter.
So there's no contradiction between saying that a radiation pattern is a gaussian beam and saying that it diverges with a known angle.