In the general case (i.e. not specific to Stellaris), software is software, the bootloader is software, your application is software, the processor cannot tell the difference so quite evidently a bootloader is not required, the software that runs at reset could as easily be your software.
However the obvious benefit of a bootloader is the ability to apply in-field updates without connecting special equipment; you might regret loosing that capability.
Some chips (again generally, check your data sheet for Stelaris specifics) have a bootloader in mask ROM rather than Flash and you cannot delete or overwrite that, but usually configuration pins can be set to select the boot behaviour in order to by-pass the bootloader for example.