Hello,
The dielectric constant is a material property, so when you use FR4 as a substrate (cheap, good available, every one can process it), you get about 4 as relative dielectric constant.
When you use a Teflon substrate (better and more consistent properties for microwave, but more expensive an mechanically weak), you get about 2.2 as relative dielectric constant. When you decide to use ceramic (GPS patch antennas), you get other electrical and mechanical properties.
For good antenna operation, it isn't needed as you can also make a patch in air where you only have a support in the middle (where the E-field strength has its minimum for a half wave resonant patch). When you put dielectric under the patch, you have to reduce the size to get it in resonance again.
Small detail: the additional displacement current that you get in the dielectric, does radiate (as it is real vibration of charge), so the dielectric does affect the radiation pattern under zero elevation (for a horizontal patch).