No, one is not required according to Jewish law to bury 6 feet underground. In fact, there is no source for any minimum depth in Halacha, although there are different customs ranging from 10 Tefachim [80 cm] to 1.7 meters [slightly less than 6 feet, Six feet is about 1.83 meters, which equals 183 centimeters]. Interestingly, I have not found any source which mentions a depth of 6 feet and the closest is the custom of the Kabbalists for it to be 1.7 m [5.58 feet]. To also note that the feet measurement system did not exist in Jewish literature which follows Amos, so the notion that it would say somewhere to bury 6 feet into the ground is preposterous. At most, it would state and Ama or meter measurement that may add up to that amount. So here then does it come from? “Six feet under” comes from the traditional depth at which many graves were dug across various cultures and religions, including Christian tradition to ensure bodies were well-covered and to prevent animals from uncovering them and help contain disease. The phrase became a metaphor for death or burial over time. However, it is not a Halachic requirement.

See Gesher Hachaim 16:4 who suggests the following depths 1) 1.25 m and more as the standard practice; 2) Ten Tefachim [80 cm] and more in order so it be considered its own domain; 3) Jerusalem custom for it to be between 1.3 – 1.4 meters. 4) Kabbalists who instructed it to be 1.7 m.; Darkei Chesed 17:12; See Arizal in Shaar Hagilgulim Hakdama 23 that the grave must be dug deep, in order for there to be room for the shaking process of Chibut Hakever to transpire.