The exchange began after the Texas senator took a jab at Trevor Noah, the current host of The Daily Show, who took over from Stewart in 2015 after 16 years.
Jon Stewart has hit back at Ted Cruz after he attempted to mock The Daily Show on social media.
Cruz was reacting to a Daily Show segment in which Noah commented on the 2020 census.
The US study impacts how many seats each state has in Congress, with Texas is poised to gain two as a result of the latest census.