Any medium will have dispersion. Free space itself has linear dispersion (phase is linear with frequency). A previous reply mentioned phase [delay, or change] is related to a time delay. Yes, from a signals perspective, this is true. You are viewing it as propagation through a medium (of a given distance or time delay). Note, there is only a single relationship between phase and time (or distance) if the phase slope (versus frequency) is linear, as in free space. This is equivalent to a constant phase velocity. This is also equivalent to saying that the group velocity is constant or that group velocity dispersion (GVD) is zero. Usually, a medium is dispersive, in that phase slope isn't quite linear, and so a pulse will spread (in time, or distance) as it propagates.
There is usually some attenuation as well, and one can lump all these effects into a complex propagation "constant" or wavevector. But, I think you can figure that part on your own. Working with chirped pulses, you will likely be looking at higher-order dispersion terms (GVD is just the first) so, there are good resources out there...
You are really asking a more fundamental question. What is the phase of a [gaussian] pulse? The time and frequency domains are simply related through the Fourier transform. The transform of a gaussian (in time) is also a gaussian in frequency. Just look at Mathworld. Note, however, that the complex part of the transform integrates to zero. This generally true of all even waveforms. However, one usually doesn't have a gaussian (in time) centered at zero time, but instead at some finite t0 time. If you do that, you will find a linear phase offset (linear with frequency). This is equivalent to a port offset on a network analyzer, or simply a propagation distance in physics, or an application of the Fourier transform "shifting theorem". Multiplying the wave by exp(i*2pi*f*t0) will shift the waveform in time by t0, and add a linear slope offset to the phase spectrum.
Even more fundamental: What is the "phase" of your chirped gaussian pulse? Yeah, one can still have a gaussian amplitude ("envelope") and any phase spectrum you wish. The time domain waveform is then the inverse transform. One talks about a "transform-limited" pulse when the pulse width is as narrow as possible in time for the spectral content. Any kind of dispersion will broaden the pulse (and produce some chirp). The simplest kind of dispersion thus has only GVD with no higher order (cubic, quadratic, etc). That is, the phase versus frequency is a quadratic polynomial. The linear term can be zero if the pulse is centered at time=0, or it can have a finite linear term if the pulse is offset in time. But the second-order term will produce a waveform with the wave skewed toward one side of the envelope. Note, the amplitude is still gaussian, but the real component is lopsided.
This discussion is analogous to applying filter functions, either in the time domain or frequency domain. Just be careful with phase. Many folks ignore it altogether, only looking at the amplitude portion. The phase portion is essential to preserving the transforms and shape in the time domain. Be careful with the format of your data. Any real time waveform must have a hermitian spectrum. You need to be careful with digitized data, and know what your canned package is doing to the format of the data. Typically, the data is in a format analogous to a signed integer, where negative frequency wraps around to zero at the end of the waveform, e.g. FF is -1 for an 8-bit integer. Some packages truncate second half end of the data entirely (since it must be hermitian and "duplicates" the first half)...
I would suggest some reading on Fourier transforms, such as Ron Bracewell.